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Theorem canthp1lem2 9354
 Description: Lemma for canthp1 9355. (Contributed by Mario Carneiro, 18-May-2015.)
Hypotheses
Ref Expression
canthp1lem2.1 (𝜑 → 1𝑜𝐴)
canthp1lem2.2 (𝜑𝐹:𝒫 𝐴1-1-onto→(𝐴 +𝑐 1𝑜))
canthp1lem2.3 (𝜑𝐺:((𝐴 +𝑐 1𝑜) ∖ {(𝐹𝐴)})–1-1-onto𝐴)
canthp1lem2.4 𝐻 = ((𝐺𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))
canthp1lem2.5 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐻‘(𝑟 “ {𝑦})) = 𝑦))}
canthp1lem2.6 𝐵 = dom 𝑊
Assertion
Ref Expression
canthp1lem2 ¬ 𝜑
Distinct variable groups:   𝑥,𝑟,𝑦,𝐴   𝐵,𝑟,𝑥,𝑦   𝐻,𝑟,𝑥,𝑦   𝜑,𝑟,𝑥,𝑦   𝑊,𝑟,𝑥,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑟)   𝐺(𝑥,𝑦,𝑟)

Proof of Theorem canthp1lem2
StepHypRef Expression
1 canthp1lem2.1 . . . . . 6 (𝜑 → 1𝑜𝐴)
2 relsdom 7848 . . . . . . 7 Rel ≺
32brrelex2i 5083 . . . . . 6 (1𝑜𝐴𝐴 ∈ V)
41, 3syl 17 . . . . 5 (𝜑𝐴 ∈ V)
5 pwexg 4776 . . . . 5 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
64, 5syl 17 . . . 4 (𝜑 → 𝒫 𝐴 ∈ V)
7 canthp1lem2.2 . . . 4 (𝜑𝐹:𝒫 𝐴1-1-onto→(𝐴 +𝑐 1𝑜))
8 f1oeng 7860 . . . 4 ((𝒫 𝐴 ∈ V ∧ 𝐹:𝒫 𝐴1-1-onto→(𝐴 +𝑐 1𝑜)) → 𝒫 𝐴 ≈ (𝐴 +𝑐 1𝑜))
96, 7, 8syl2anc 691 . . 3 (𝜑 → 𝒫 𝐴 ≈ (𝐴 +𝑐 1𝑜))
10 ensym 7891 . . 3 (𝒫 𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 +𝑐 1𝑜) ≈ 𝒫 𝐴)
119, 10syl 17 . 2 (𝜑 → (𝐴 +𝑐 1𝑜) ≈ 𝒫 𝐴)
12 canth2g 7999 . . . . . . . . . . 11 (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
134, 12syl 17 . . . . . . . . . 10 (𝜑𝐴 ≺ 𝒫 𝐴)
14 sdomen2 7990 . . . . . . . . . . 11 (𝒫 𝐴 ≈ (𝐴 +𝑐 1𝑜) → (𝐴 ≺ 𝒫 𝐴𝐴 ≺ (𝐴 +𝑐 1𝑜)))
159, 14syl 17 . . . . . . . . . 10 (𝜑 → (𝐴 ≺ 𝒫 𝐴𝐴 ≺ (𝐴 +𝑐 1𝑜)))
1613, 15mpbid 221 . . . . . . . . 9 (𝜑𝐴 ≺ (𝐴 +𝑐 1𝑜))
17 sdomnen 7870 . . . . . . . . 9 (𝐴 ≺ (𝐴 +𝑐 1𝑜) → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
1816, 17syl 17 . . . . . . . 8 (𝜑 → ¬ 𝐴 ≈ (𝐴 +𝑐 1𝑜))
19 omelon 8426 . . . . . . . . . . . 12 ω ∈ On
20 onenon 8658 . . . . . . . . . . . 12 (ω ∈ On → ω ∈ dom card)
2119, 20ax-mp 5 . . . . . . . . . . 11 ω ∈ dom card
22 canthp1lem2.3 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐺:((𝐴 +𝑐 1𝑜) ∖ {(𝐹𝐴)})–1-1-onto𝐴)
23 dff1o3 6056 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:𝒫 𝐴1-1-onto→(𝐴 +𝑐 1𝑜) ↔ (𝐹:𝒫 𝐴onto→(𝐴 +𝑐 1𝑜) ∧ Fun 𝐹))
2423simprbi 479 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:𝒫 𝐴1-1-onto→(𝐴 +𝑐 1𝑜) → Fun 𝐹)
257, 24syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → Fun 𝐹)
26 f1ofo 6057 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:𝒫 𝐴1-1-onto→(𝐴 +𝑐 1𝑜) → 𝐹:𝒫 𝐴onto→(𝐴 +𝑐 1𝑜))
277, 26syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐹:𝒫 𝐴onto→(𝐴 +𝑐 1𝑜))
28 f1ofn 6051 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:𝒫 𝐴1-1-onto→(𝐴 +𝑐 1𝑜) → 𝐹 Fn 𝒫 𝐴)
29 fnresdm 5914 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹 Fn 𝒫 𝐴 → (𝐹 ↾ 𝒫 𝐴) = 𝐹)
30 foeq1 6024 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 ↾ 𝒫 𝐴) = 𝐹 → ((𝐹 ↾ 𝒫 𝐴):𝒫 𝐴onto→(𝐴 +𝑐 1𝑜) ↔ 𝐹:𝒫 𝐴onto→(𝐴 +𝑐 1𝑜)))
317, 28, 29, 304syl 19 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝐹 ↾ 𝒫 𝐴):𝒫 𝐴onto→(𝐴 +𝑐 1𝑜) ↔ 𝐹:𝒫 𝐴onto→(𝐴 +𝑐 1𝑜)))
3227, 31mpbird 246 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐹 ↾ 𝒫 𝐴):𝒫 𝐴onto→(𝐴 +𝑐 1𝑜))
33 fvex 6113 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹𝐴) ∈ V
34 f1osng 6089 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ V ∧ (𝐹𝐴) ∈ V) → {⟨𝐴, (𝐹𝐴)⟩}:{𝐴}–1-1-onto→{(𝐹𝐴)})
354, 33, 34sylancl 693 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → {⟨𝐴, (𝐹𝐴)⟩}:{𝐴}–1-1-onto→{(𝐹𝐴)})
367, 28syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐹 Fn 𝒫 𝐴)
37 pwidg 4121 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
384, 37syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐴 ∈ 𝒫 𝐴)
39 fnressn 6330 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐹 Fn 𝒫 𝐴𝐴 ∈ 𝒫 𝐴) → (𝐹 ↾ {𝐴}) = {⟨𝐴, (𝐹𝐴)⟩})
4036, 38, 39syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐹 ↾ {𝐴}) = {⟨𝐴, (𝐹𝐴)⟩})
41 f1oeq1 6040 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹 ↾ {𝐴}) = {⟨𝐴, (𝐹𝐴)⟩} → ((𝐹 ↾ {𝐴}):{𝐴}–1-1-onto→{(𝐹𝐴)} ↔ {⟨𝐴, (𝐹𝐴)⟩}:{𝐴}–1-1-onto→{(𝐹𝐴)}))
4240, 41syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝐹 ↾ {𝐴}):{𝐴}–1-1-onto→{(𝐹𝐴)} ↔ {⟨𝐴, (𝐹𝐴)⟩}:{𝐴}–1-1-onto→{(𝐹𝐴)}))
4335, 42mpbird 246 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐹 ↾ {𝐴}):{𝐴}–1-1-onto→{(𝐹𝐴)})
44 f1ofo 6057 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 ↾ {𝐴}):{𝐴}–1-1-onto→{(𝐹𝐴)} → (𝐹 ↾ {𝐴}):{𝐴}–onto→{(𝐹𝐴)})
4543, 44syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐹 ↾ {𝐴}):{𝐴}–onto→{(𝐹𝐴)})
46 resdif 6070 . . . . . . . . . . . . . . . . . . . . 21 ((Fun 𝐹 ∧ (𝐹 ↾ 𝒫 𝐴):𝒫 𝐴onto→(𝐴 +𝑐 1𝑜) ∧ (𝐹 ↾ {𝐴}):{𝐴}–onto→{(𝐹𝐴)}) → (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→((𝐴 +𝑐 1𝑜) ∖ {(𝐹𝐴)}))
4725, 32, 45, 46syl3anc 1318 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→((𝐴 +𝑐 1𝑜) ∖ {(𝐹𝐴)}))
48 f1oco 6072 . . . . . . . . . . . . . . . . . . . 20 ((𝐺:((𝐴 +𝑐 1𝑜) ∖ {(𝐹𝐴)})–1-1-onto𝐴 ∧ (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto→((𝐴 +𝑐 1𝑜) ∖ {(𝐹𝐴)})) → (𝐺 ∘ (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴}))):(𝒫 𝐴 ∖ {𝐴})–1-1-onto𝐴)
4922, 47, 48syl2anc 691 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐺 ∘ (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴}))):(𝒫 𝐴 ∖ {𝐴})–1-1-onto𝐴)
50 resco 5556 . . . . . . . . . . . . . . . . . . . 20 ((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) = (𝐺 ∘ (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴})))
51 f1oeq1 6040 . . . . . . . . . . . . . . . . . . . 20 (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) = (𝐺 ∘ (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴}))) → (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto𝐴 ↔ (𝐺 ∘ (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴}))):(𝒫 𝐴 ∖ {𝐴})–1-1-onto𝐴))
5250, 51ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto𝐴 ↔ (𝐺 ∘ (𝐹 ↾ (𝒫 𝐴 ∖ {𝐴}))):(𝒫 𝐴 ∖ {𝐴})–1-1-onto𝐴)
5349, 52sylibr 223 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto𝐴)
54 f1of 6050 . . . . . . . . . . . . . . . . . 18 (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto𝐴 → ((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})⟶𝐴)
5553, 54syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})⟶𝐴)
56 0elpw 4760 . . . . . . . . . . . . . . . . . . . . 21 ∅ ∈ 𝒫 𝐴
5756a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 = 𝐴) → ∅ ∈ 𝒫 𝐴)
58 sdom0 7977 . . . . . . . . . . . . . . . . . . . . . . . 24 ¬ 1𝑜 ≺ ∅
59 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . . 24 (∅ = 𝐴 → (1𝑜 ≺ ∅ ↔ 1𝑜𝐴))
6058, 59mtbii 315 . . . . . . . . . . . . . . . . . . . . . . 23 (∅ = 𝐴 → ¬ 1𝑜𝐴)
6160necon2ai 2811 . . . . . . . . . . . . . . . . . . . . . 22 (1𝑜𝐴 → ∅ ≠ 𝐴)
621, 61syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ∅ ≠ 𝐴)
6362ad2antrr 758 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 = 𝐴) → ∅ ≠ 𝐴)
64 eldifsn 4260 . . . . . . . . . . . . . . . . . . . 20 (∅ ∈ (𝒫 𝐴 ∖ {𝐴}) ↔ (∅ ∈ 𝒫 𝐴 ∧ ∅ ≠ 𝐴))
6557, 63, 64sylanbrc 695 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ 𝒫 𝐴) ∧ 𝑥 = 𝐴) → ∅ ∈ (𝒫 𝐴 ∖ {𝐴}))
66 simplr 788 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ 𝒫 𝐴) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ∈ 𝒫 𝐴)
67 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ 𝒫 𝐴) ∧ ¬ 𝑥 = 𝐴) → ¬ 𝑥 = 𝐴)
6867neqned 2789 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ 𝒫 𝐴) ∧ ¬ 𝑥 = 𝐴) → 𝑥𝐴)
69 eldifsn 4260 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝐴 ∖ {𝐴}) ↔ (𝑥 ∈ 𝒫 𝐴𝑥𝐴))
7066, 68, 69sylanbrc 695 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ 𝒫 𝐴) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ∈ (𝒫 𝐴 ∖ {𝐴}))
7165, 70ifclda 4070 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ 𝒫 𝐴) → if(𝑥 = 𝐴, ∅, 𝑥) ∈ (𝒫 𝐴 ∖ {𝐴}))
72 eqid 2610 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)) = (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))
7371, 72fmptd 6292 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)):𝒫 𝐴⟶(𝒫 𝐴 ∖ {𝐴}))
74 fco 5971 . . . . . . . . . . . . . . . . 17 ((((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})⟶𝐴 ∧ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)):𝒫 𝐴⟶(𝒫 𝐴 ∖ {𝐴})) → (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))):𝒫 𝐴𝐴)
7555, 73, 74syl2anc 691 . . . . . . . . . . . . . . . 16 (𝜑 → (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))):𝒫 𝐴𝐴)
76 frn 5966 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)):𝒫 𝐴⟶(𝒫 𝐴 ∖ {𝐴}) → ran (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)) ⊆ (𝒫 𝐴 ∖ {𝐴}))
7773, 76syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ran (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)) ⊆ (𝒫 𝐴 ∖ {𝐴}))
78 cores 5555 . . . . . . . . . . . . . . . . . . 19 (ran (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)) ⊆ (𝒫 𝐴 ∖ {𝐴}) → (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))) = ((𝐺𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))))
7977, 78syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))) = ((𝐺𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))))
80 canthp1lem2.4 . . . . . . . . . . . . . . . . . 18 𝐻 = ((𝐺𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))
8179, 80syl6eqr 2662 . . . . . . . . . . . . . . . . 17 (𝜑 → (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))) = 𝐻)
8281feq1d 5943 . . . . . . . . . . . . . . . 16 (𝜑 → ((((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))):𝒫 𝐴𝐴𝐻:𝒫 𝐴𝐴))
8375, 82mpbid 221 . . . . . . . . . . . . . . 15 (𝜑𝐻:𝒫 𝐴𝐴)
84 inss1 3795 . . . . . . . . . . . . . . . 16 (𝒫 𝐴 ∩ dom card) ⊆ 𝒫 𝐴
8584a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (𝒫 𝐴 ∩ dom card) ⊆ 𝒫 𝐴)
86 canthp1lem2.5 . . . . . . . . . . . . . . . 16 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐻‘(𝑟 “ {𝑦})) = 𝑦))}
87 canthp1lem2.6 . . . . . . . . . . . . . . . 16 𝐵 = dom 𝑊
88 eqid 2610 . . . . . . . . . . . . . . . 16 ((𝑊𝐵) “ {(𝐻𝐵)}) = ((𝑊𝐵) “ {(𝐻𝐵)})
8986, 87, 88canth4 9348 . . . . . . . . . . . . . . 15 ((𝐴 ∈ V ∧ 𝐻:𝒫 𝐴𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝒫 𝐴) → (𝐵𝐴 ∧ ((𝑊𝐵) “ {(𝐻𝐵)}) ⊊ 𝐵 ∧ (𝐻𝐵) = (𝐻‘((𝑊𝐵) “ {(𝐻𝐵)}))))
904, 83, 85, 89syl3anc 1318 . . . . . . . . . . . . . 14 (𝜑 → (𝐵𝐴 ∧ ((𝑊𝐵) “ {(𝐻𝐵)}) ⊊ 𝐵 ∧ (𝐻𝐵) = (𝐻‘((𝑊𝐵) “ {(𝐻𝐵)}))))
9190simp1d 1066 . . . . . . . . . . . . 13 (𝜑𝐵𝐴)
9290simp2d 1067 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑊𝐵) “ {(𝐻𝐵)}) ⊊ 𝐵)
9392pssned 3667 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑊𝐵) “ {(𝐻𝐵)}) ≠ 𝐵)
9493necomd 2837 . . . . . . . . . . . . . . 15 (𝜑𝐵 ≠ ((𝑊𝐵) “ {(𝐻𝐵)}))
9590simp3d 1068 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐻𝐵) = (𝐻‘((𝑊𝐵) “ {(𝐻𝐵)})))
9680fveq1i 6104 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐻𝐵) = (((𝐺𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘𝐵)
9780fveq1i 6104 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐻‘((𝑊𝐵) “ {(𝐻𝐵)})) = (((𝐺𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘((𝑊𝐵) “ {(𝐻𝐵)}))
9895, 96, 973eqtr3g 2667 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((𝐺𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘𝐵) = (((𝐺𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘((𝑊𝐵) “ {(𝐻𝐵)})))
99 elpw2g 4754 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ V → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
1004, 99syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
10191, 100mpbird 246 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐵 ∈ 𝒫 𝐴)
102 fvco3 6185 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)):𝒫 𝐴⟶(𝒫 𝐴 ∖ {𝐴}) ∧ 𝐵 ∈ 𝒫 𝐴) → (((𝐺𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘𝐵) = ((𝐺𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵)))
10373, 101, 102syl2anc 691 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((𝐺𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘𝐵) = ((𝐺𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵)))
10492pssssd 3666 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((𝑊𝐵) “ {(𝐻𝐵)}) ⊆ 𝐵)
105104, 91sstrd 3578 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((𝑊𝐵) “ {(𝐻𝐵)}) ⊆ 𝐴)
106 elpw2g 4754 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ V → (((𝑊𝐵) “ {(𝐻𝐵)}) ∈ 𝒫 𝐴 ↔ ((𝑊𝐵) “ {(𝐻𝐵)}) ⊆ 𝐴))
1074, 106syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (((𝑊𝐵) “ {(𝐻𝐵)}) ∈ 𝒫 𝐴 ↔ ((𝑊𝐵) “ {(𝐻𝐵)}) ⊆ 𝐴))
108105, 107mpbird 246 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑊𝐵) “ {(𝐻𝐵)}) ∈ 𝒫 𝐴)
109 fvco3 6185 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)):𝒫 𝐴⟶(𝒫 𝐴 ∖ {𝐴}) ∧ ((𝑊𝐵) “ {(𝐻𝐵)}) ∈ 𝒫 𝐴) → (((𝐺𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘((𝑊𝐵) “ {(𝐻𝐵)})) = ((𝐺𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘((𝑊𝐵) “ {(𝐻𝐵)}))))
11073, 108, 109syl2anc 691 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((𝐺𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))‘((𝑊𝐵) “ {(𝐻𝐵)})) = ((𝐺𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘((𝑊𝐵) “ {(𝐻𝐵)}))))
11198, 103, 1103eqtr3d 2652 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝐺𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵)) = ((𝐺𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘((𝑊𝐵) “ {(𝐻𝐵)}))))
112111adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝐵𝐴) → ((𝐺𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵)) = ((𝐺𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘((𝑊𝐵) “ {(𝐻𝐵)}))))
113 ifcl 4080 . . . . . . . . . . . . . . . . . . . . . . . 24 ((∅ ∈ 𝒫 𝐴𝐵 ∈ 𝒫 𝐴) → if(𝐵 = 𝐴, ∅, 𝐵) ∈ 𝒫 𝐴)
11456, 101, 113sylancr 694 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → if(𝐵 = 𝐴, ∅, 𝐵) ∈ 𝒫 𝐴)
115 eqeq1 2614 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝐵 → (𝑥 = 𝐴𝐵 = 𝐴))
116 id 22 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝐵𝑥 = 𝐵)
117115, 116ifbieq2d 4061 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝐵 → if(𝑥 = 𝐴, ∅, 𝑥) = if(𝐵 = 𝐴, ∅, 𝐵))
118117, 72fvmptg 6189 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵 ∈ 𝒫 𝐴 ∧ if(𝐵 = 𝐴, ∅, 𝐵) ∈ 𝒫 𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵) = if(𝐵 = 𝐴, ∅, 𝐵))
119101, 114, 118syl2anc 691 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵) = if(𝐵 = 𝐴, ∅, 𝐵))
120 pssne 3665 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐵𝐴𝐵𝐴)
121120neneqd 2787 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐵𝐴 → ¬ 𝐵 = 𝐴)
122121iffalsed 4047 . . . . . . . . . . . . . . . . . . . . . 22 (𝐵𝐴 → if(𝐵 = 𝐴, ∅, 𝐵) = 𝐵)
123119, 122sylan9eq 2664 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝐵𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵) = 𝐵)
124123fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝐵𝐴) → ((𝐺𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘𝐵)) = ((𝐺𝐹)‘𝐵))
125 ifcl 4080 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((∅ ∈ 𝒫 𝐴 ∧ ((𝑊𝐵) “ {(𝐻𝐵)}) ∈ 𝒫 𝐴) → if(((𝑊𝐵) “ {(𝐻𝐵)}) = 𝐴, ∅, ((𝑊𝐵) “ {(𝐻𝐵)})) ∈ 𝒫 𝐴)
12656, 108, 125sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → if(((𝑊𝐵) “ {(𝐻𝐵)}) = 𝐴, ∅, ((𝑊𝐵) “ {(𝐻𝐵)})) ∈ 𝒫 𝐴)
127 eqeq1 2614 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ((𝑊𝐵) “ {(𝐻𝐵)}) → (𝑥 = 𝐴 ↔ ((𝑊𝐵) “ {(𝐻𝐵)}) = 𝐴))
128 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ((𝑊𝐵) “ {(𝐻𝐵)}) → 𝑥 = ((𝑊𝐵) “ {(𝐻𝐵)}))
129127, 128ifbieq2d 4061 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ((𝑊𝐵) “ {(𝐻𝐵)}) → if(𝑥 = 𝐴, ∅, 𝑥) = if(((𝑊𝐵) “ {(𝐻𝐵)}) = 𝐴, ∅, ((𝑊𝐵) “ {(𝐻𝐵)})))
130129, 72fvmptg 6189 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑊𝐵) “ {(𝐻𝐵)}) ∈ 𝒫 𝐴 ∧ if(((𝑊𝐵) “ {(𝐻𝐵)}) = 𝐴, ∅, ((𝑊𝐵) “ {(𝐻𝐵)})) ∈ 𝒫 𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘((𝑊𝐵) “ {(𝐻𝐵)})) = if(((𝑊𝐵) “ {(𝐻𝐵)}) = 𝐴, ∅, ((𝑊𝐵) “ {(𝐻𝐵)})))
131108, 126, 130syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘((𝑊𝐵) “ {(𝐻𝐵)})) = if(((𝑊𝐵) “ {(𝐻𝐵)}) = 𝐴, ∅, ((𝑊𝐵) “ {(𝐻𝐵)})))
132131adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝐵𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘((𝑊𝐵) “ {(𝐻𝐵)})) = if(((𝑊𝐵) “ {(𝐻𝐵)}) = 𝐴, ∅, ((𝑊𝐵) “ {(𝐻𝐵)})))
133 sspsstr 3674 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑊𝐵) “ {(𝐻𝐵)}) ⊆ 𝐵𝐵𝐴) → ((𝑊𝐵) “ {(𝐻𝐵)}) ⊊ 𝐴)
134104, 133sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝐵𝐴) → ((𝑊𝐵) “ {(𝐻𝐵)}) ⊊ 𝐴)
135134pssned 3667 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐵𝐴) → ((𝑊𝐵) “ {(𝐻𝐵)}) ≠ 𝐴)
136135neneqd 2787 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐵𝐴) → ¬ ((𝑊𝐵) “ {(𝐻𝐵)}) = 𝐴)
137136iffalsed 4047 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝐵𝐴) → if(((𝑊𝐵) “ {(𝐻𝐵)}) = 𝐴, ∅, ((𝑊𝐵) “ {(𝐻𝐵)})) = ((𝑊𝐵) “ {(𝐻𝐵)}))
138132, 137eqtrd 2644 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝐵𝐴) → ((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘((𝑊𝐵) “ {(𝐻𝐵)})) = ((𝑊𝐵) “ {(𝐻𝐵)}))
139138fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝐵𝐴) → ((𝐺𝐹)‘((𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))‘((𝑊𝐵) “ {(𝐻𝐵)}))) = ((𝐺𝐹)‘((𝑊𝐵) “ {(𝐻𝐵)})))
140112, 124, 1393eqtr3d 2652 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝐵𝐴) → ((𝐺𝐹)‘𝐵) = ((𝐺𝐹)‘((𝑊𝐵) “ {(𝐻𝐵)})))
141101, 120anim12i 588 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝐵𝐴) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
142 eldifsn 4260 . . . . . . . . . . . . . . . . . . . . 21 (𝐵 ∈ (𝒫 𝐴 ∖ {𝐴}) ↔ (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
143141, 142sylibr 223 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝐵𝐴) → 𝐵 ∈ (𝒫 𝐴 ∖ {𝐴}))
144 fvres 6117 . . . . . . . . . . . . . . . . . . . 20 (𝐵 ∈ (𝒫 𝐴 ∖ {𝐴}) → (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘𝐵) = ((𝐺𝐹)‘𝐵))
145143, 144syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝐵𝐴) → (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘𝐵) = ((𝐺𝐹)‘𝐵))
146108adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝐵𝐴) → ((𝑊𝐵) “ {(𝐻𝐵)}) ∈ 𝒫 𝐴)
147 eldifsn 4260 . . . . . . . . . . . . . . . . . . . . 21 (((𝑊𝐵) “ {(𝐻𝐵)}) ∈ (𝒫 𝐴 ∖ {𝐴}) ↔ (((𝑊𝐵) “ {(𝐻𝐵)}) ∈ 𝒫 𝐴 ∧ ((𝑊𝐵) “ {(𝐻𝐵)}) ≠ 𝐴))
148146, 135, 147sylanbrc 695 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝐵𝐴) → ((𝑊𝐵) “ {(𝐻𝐵)}) ∈ (𝒫 𝐴 ∖ {𝐴}))
149 fvres 6117 . . . . . . . . . . . . . . . . . . . 20 (((𝑊𝐵) “ {(𝐻𝐵)}) ∈ (𝒫 𝐴 ∖ {𝐴}) → (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘((𝑊𝐵) “ {(𝐻𝐵)})) = ((𝐺𝐹)‘((𝑊𝐵) “ {(𝐻𝐵)})))
150148, 149syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝐵𝐴) → (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘((𝑊𝐵) “ {(𝐻𝐵)})) = ((𝐺𝐹)‘((𝑊𝐵) “ {(𝐻𝐵)})))
151140, 145, 1503eqtr4d 2654 . . . . . . . . . . . . . . . . . 18 ((𝜑𝐵𝐴) → (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘𝐵) = (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘((𝑊𝐵) “ {(𝐻𝐵)})))
152 f1of1 6049 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1-onto𝐴 → ((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1𝐴)
15353, 152syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1𝐴)
154153adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝐵𝐴) → ((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1𝐴)
155 f1fveq 6420 . . . . . . . . . . . . . . . . . . 19 ((((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴})):(𝒫 𝐴 ∖ {𝐴})–1-1𝐴 ∧ (𝐵 ∈ (𝒫 𝐴 ∖ {𝐴}) ∧ ((𝑊𝐵) “ {(𝐻𝐵)}) ∈ (𝒫 𝐴 ∖ {𝐴}))) → ((((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘𝐵) = (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘((𝑊𝐵) “ {(𝐻𝐵)})) ↔ 𝐵 = ((𝑊𝐵) “ {(𝐻𝐵)})))
156154, 143, 148, 155syl12anc 1316 . . . . . . . . . . . . . . . . . 18 ((𝜑𝐵𝐴) → ((((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘𝐵) = (((𝐺𝐹) ↾ (𝒫 𝐴 ∖ {𝐴}))‘((𝑊𝐵) “ {(𝐻𝐵)})) ↔ 𝐵 = ((𝑊𝐵) “ {(𝐻𝐵)})))
157151, 156mpbid 221 . . . . . . . . . . . . . . . . 17 ((𝜑𝐵𝐴) → 𝐵 = ((𝑊𝐵) “ {(𝐻𝐵)}))
158157ex 449 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐵𝐴𝐵 = ((𝑊𝐵) “ {(𝐻𝐵)})))
159158necon3ad 2795 . . . . . . . . . . . . . . 15 (𝜑 → (𝐵 ≠ ((𝑊𝐵) “ {(𝐻𝐵)}) → ¬ 𝐵𝐴))
16094, 159mpd 15 . . . . . . . . . . . . . 14 (𝜑 → ¬ 𝐵𝐴)
161 npss 3679 . . . . . . . . . . . . . 14 𝐵𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))
162160, 161sylib 207 . . . . . . . . . . . . 13 (𝜑 → (𝐵𝐴𝐵 = 𝐴))
16391, 162mpd 15 . . . . . . . . . . . 12 (𝜑𝐵 = 𝐴)
164 eqid 2610 . . . . . . . . . . . . . . . . . . . 20 𝐵 = 𝐵
165 eqid 2610 . . . . . . . . . . . . . . . . . . . 20 (𝑊𝐵) = (𝑊𝐵)
166164, 165pm3.2i 470 . . . . . . . . . . . . . . . . . . 19 (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))
16784sseli 3564 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝐴 ∩ dom card) → 𝑥 ∈ 𝒫 𝐴)
168 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . 21 ((𝐻:𝒫 𝐴𝐴𝑥 ∈ 𝒫 𝐴) → (𝐻𝑥) ∈ 𝐴)
16983, 167, 168syl2an 493 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐻𝑥) ∈ 𝐴)
17086, 4, 169, 87fpwwe 9347 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐵𝑊(𝑊𝐵) ∧ (𝐻𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))))
171166, 170mpbiri 247 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐵𝑊(𝑊𝐵) ∧ (𝐻𝐵) ∈ 𝐵))
172171simpld 474 . . . . . . . . . . . . . . . . 17 (𝜑𝐵𝑊(𝑊𝐵))
17386, 4fpwwelem 9346 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐵𝑊(𝑊𝐵) ↔ ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐻‘((𝑊𝐵) “ {𝑦})) = 𝑦))))
174172, 173mpbid 221 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐻‘((𝑊𝐵) “ {𝑦})) = 𝑦)))
175174simprd 478 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐻‘((𝑊𝐵) “ {𝑦})) = 𝑦))
176175simpld 474 . . . . . . . . . . . . . 14 (𝜑 → (𝑊𝐵) We 𝐵)
177 fvex 6113 . . . . . . . . . . . . . . 15 (𝑊𝐵) ∈ V
178 weeq1 5026 . . . . . . . . . . . . . . 15 (𝑟 = (𝑊𝐵) → (𝑟 We 𝐵 ↔ (𝑊𝐵) We 𝐵))
179177, 178spcev 3273 . . . . . . . . . . . . . 14 ((𝑊𝐵) We 𝐵 → ∃𝑟 𝑟 We 𝐵)
180176, 179syl 17 . . . . . . . . . . . . 13 (𝜑 → ∃𝑟 𝑟 We 𝐵)
181 ween 8741 . . . . . . . . . . . . 13 (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵)
182180, 181sylibr 223 . . . . . . . . . . . 12 (𝜑𝐵 ∈ dom card)
183163, 182eqeltrrd 2689 . . . . . . . . . . 11 (𝜑𝐴 ∈ dom card)
184 domtri2 8698 . . . . . . . . . . 11 ((ω ∈ dom card ∧ 𝐴 ∈ dom card) → (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω))
18521, 183, 184sylancr 694 . . . . . . . . . 10 (𝜑 → (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω))
186 infcda1 8898 . . . . . . . . . 10 (ω ≼ 𝐴 → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
187185, 186syl6bir 243 . . . . . . . . 9 (𝜑 → (¬ 𝐴 ≺ ω → (𝐴 +𝑐 1𝑜) ≈ 𝐴))
188 ensym 7891 . . . . . . . . 9 ((𝐴 +𝑐 1𝑜) ≈ 𝐴𝐴 ≈ (𝐴 +𝑐 1𝑜))
189187, 188syl6 34 . . . . . . . 8 (𝜑 → (¬ 𝐴 ≺ ω → 𝐴 ≈ (𝐴 +𝑐 1𝑜)))
19018, 189mt3d 139 . . . . . . 7 (𝜑𝐴 ≺ ω)
191 2onn 7607 . . . . . . . 8 2𝑜 ∈ ω
192 nnsdom 8434 . . . . . . . 8 (2𝑜 ∈ ω → 2𝑜 ≺ ω)
193191, 192ax-mp 5 . . . . . . 7 2𝑜 ≺ ω
194 cdafi 8895 . . . . . . 7 ((𝐴 ≺ ω ∧ 2𝑜 ≺ ω) → (𝐴 +𝑐 2𝑜) ≺ ω)
195190, 193, 194sylancl 693 . . . . . 6 (𝜑 → (𝐴 +𝑐 2𝑜) ≺ ω)
196 isfinite 8432 . . . . . 6 ((𝐴 +𝑐 2𝑜) ∈ Fin ↔ (𝐴 +𝑐 2𝑜) ≺ ω)
197195, 196sylibr 223 . . . . 5 (𝜑 → (𝐴 +𝑐 2𝑜) ∈ Fin)
198 sssucid 5719 . . . . . . . . . 10 1𝑜 ⊆ suc 1𝑜
199 df-2o 7448 . . . . . . . . . 10 2𝑜 = suc 1𝑜
200198, 199sseqtr4i 3601 . . . . . . . . 9 1𝑜 ⊆ 2𝑜
201 xpss1 5151 . . . . . . . . 9 (1𝑜 ⊆ 2𝑜 → (1𝑜 × {1𝑜}) ⊆ (2𝑜 × {1𝑜}))
202200, 201ax-mp 5 . . . . . . . 8 (1𝑜 × {1𝑜}) ⊆ (2𝑜 × {1𝑜})
203 unss2 3746 . . . . . . . 8 ((1𝑜 × {1𝑜}) ⊆ (2𝑜 × {1𝑜}) → ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊆ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})))
204202, 203mp1i 13 . . . . . . 7 (𝜑 → ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊆ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})))
205 ssun2 3739 . . . . . . . . 9 (2𝑜 × {1𝑜}) ⊆ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜}))
206 1onn 7606 . . . . . . . . . . . . 13 1𝑜 ∈ ω
207206elexi 3186 . . . . . . . . . . . 12 1𝑜 ∈ V
208207sucid 5721 . . . . . . . . . . 11 1𝑜 ∈ suc 1𝑜
209208, 199eleqtrri 2687 . . . . . . . . . 10 1𝑜 ∈ 2𝑜
210207snid 4155 . . . . . . . . . 10 1𝑜 ∈ {1𝑜}
211 opelxpi 5072 . . . . . . . . . 10 ((1𝑜 ∈ 2𝑜 ∧ 1𝑜 ∈ {1𝑜}) → ⟨1𝑜, 1𝑜⟩ ∈ (2𝑜 × {1𝑜}))
212209, 210, 211mp2an 704 . . . . . . . . 9 ⟨1𝑜, 1𝑜⟩ ∈ (2𝑜 × {1𝑜})
213205, 212sselii 3565 . . . . . . . 8 ⟨1𝑜, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜}))
214 1n0 7462 . . . . . . . . . . . 12 1𝑜 ≠ ∅
215214neii 2784 . . . . . . . . . . 11 ¬ 1𝑜 = ∅
216 opelxp2 5075 . . . . . . . . . . . 12 (⟨1𝑜, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 ∈ {∅})
217 elsni 4142 . . . . . . . . . . . 12 (1𝑜 ∈ {∅} → 1𝑜 = ∅)
218216, 217syl 17 . . . . . . . . . . 11 (⟨1𝑜, 1𝑜⟩ ∈ (𝐴 × {∅}) → 1𝑜 = ∅)
219215, 218mto 187 . . . . . . . . . 10 ¬ ⟨1𝑜, 1𝑜⟩ ∈ (𝐴 × {∅})
220 nnord 6965 . . . . . . . . . . . 12 (1𝑜 ∈ ω → Ord 1𝑜)
221 ordirr 5658 . . . . . . . . . . . 12 (Ord 1𝑜 → ¬ 1𝑜 ∈ 1𝑜)
222206, 220, 221mp2b 10 . . . . . . . . . . 11 ¬ 1𝑜 ∈ 1𝑜
223 opelxp1 5074 . . . . . . . . . . 11 (⟨1𝑜, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}) → 1𝑜 ∈ 1𝑜)
224222, 223mto 187 . . . . . . . . . 10 ¬ ⟨1𝑜, 1𝑜⟩ ∈ (1𝑜 × {1𝑜})
225219, 224pm3.2ni 895 . . . . . . . . 9 ¬ (⟨1𝑜, 1𝑜⟩ ∈ (𝐴 × {∅}) ∨ ⟨1𝑜, 1𝑜⟩ ∈ (1𝑜 × {1𝑜}))
226 elun 3715 . . . . . . . . 9 (⟨1𝑜, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ↔ (⟨1𝑜, 1𝑜⟩ ∈ (𝐴 × {∅}) ∨ ⟨1𝑜, 1𝑜⟩ ∈ (1𝑜 × {1𝑜})))
227225, 226mtbir 312 . . . . . . . 8 ¬ ⟨1𝑜, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))
228 ssnelpss 3680 . . . . . . . 8 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊆ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})) → ((⟨1𝑜, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})) ∧ ¬ ⟨1𝑜, 1𝑜⟩ ∈ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜}))) → ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊊ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜}))))
229213, 227, 228mp2ani 710 . . . . . . 7 (((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊆ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})) → ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊊ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})))
230204, 229syl 17 . . . . . 6 (𝜑 → ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊊ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})))
231 cdaval 8875 . . . . . . . 8 ((𝐴 ∈ V ∧ 1𝑜 ∈ ω) → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
2324, 206, 231sylancl 693 . . . . . . 7 (𝜑 → (𝐴 +𝑐 1𝑜) = ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})))
233 cdaval 8875 . . . . . . . 8 ((𝐴 ∈ V ∧ 2𝑜 ∈ ω) → (𝐴 +𝑐 2𝑜) = ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})))
2344, 191, 233sylancl 693 . . . . . . 7 (𝜑 → (𝐴 +𝑐 2𝑜) = ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜})))
235232, 234psseq12d 3663 . . . . . 6 (𝜑 → ((𝐴 +𝑐 1𝑜) ⊊ (𝐴 +𝑐 2𝑜) ↔ ((𝐴 × {∅}) ∪ (1𝑜 × {1𝑜})) ⊊ ((𝐴 × {∅}) ∪ (2𝑜 × {1𝑜}))))
236230, 235mpbird 246 . . . . 5 (𝜑 → (𝐴 +𝑐 1𝑜) ⊊ (𝐴 +𝑐 2𝑜))
237 php3 8031 . . . . 5 (((𝐴 +𝑐 2𝑜) ∈ Fin ∧ (𝐴 +𝑐 1𝑜) ⊊ (𝐴 +𝑐 2𝑜)) → (𝐴 +𝑐 1𝑜) ≺ (𝐴 +𝑐 2𝑜))
238197, 236, 237syl2anc 691 . . . 4 (𝜑 → (𝐴 +𝑐 1𝑜) ≺ (𝐴 +𝑐 2𝑜))
239 canthp1lem1 9353 . . . . 5 (1𝑜𝐴 → (𝐴 +𝑐 2𝑜) ≼ 𝒫 𝐴)
2401, 239syl 17 . . . 4 (𝜑 → (𝐴 +𝑐 2𝑜) ≼ 𝒫 𝐴)
241 sdomdomtr 7978 . . . 4 (((𝐴 +𝑐 1𝑜) ≺ (𝐴 +𝑐 2𝑜) ∧ (𝐴 +𝑐 2𝑜) ≼ 𝒫 𝐴) → (𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴)
242238, 240, 241syl2anc 691 . . 3 (𝜑 → (𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴)
243 sdomnen 7870 . . 3 ((𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴 → ¬ (𝐴 +𝑐 1𝑜) ≈ 𝒫 𝐴)
244242, 243syl 17 . 2 (𝜑 → ¬ (𝐴 +𝑐 1𝑜) ≈ 𝒫 𝐴)
24511, 244pm2.65i 184 1 ¬ 𝜑
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540   ⊊ wpss 3541  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  {csn 4125  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   We wwe 4996   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  Ord word 5639  Oncon0 5640  suc csuc 5642  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ωcom 6957  1𝑜c1o 7440  2𝑜c2o 7441   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  Fincfn 7841  cardccrd 8644   +𝑐 ccda 8872 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-cda 8873 This theorem is referenced by:  canthp1  9355
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