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Theorem nobnddown 31100
 Description: Any set of surreals is bounded below by a surreal with a birthday no greater than the successor of their maximum birthday. (Contributed by Scott Fenton, 10-Apr-2017.)
Assertion
Ref Expression
nobnddown ((𝐴 No 𝐴𝑉) → ∃𝑥 No (∀𝑦𝐴 𝑥 <s 𝑦 ∧ ( bday 𝑥) ⊆ suc ( bday 𝐴)))
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem nobnddown
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝐴𝑉𝐴 ∈ V)
2 1on 7454 . . . . . . 7 1𝑜 ∈ On
32elexi 3186 . . . . . 6 1𝑜 ∈ V
43prid1 4241 . . . . 5 1𝑜 ∈ {1𝑜, 2𝑜}
5 eqid 2610 . . . . 5 {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} = {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜}
64, 5nobndlem2 31092 . . . 4 ((𝐴 No 𝐴 ∈ V) → {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} ∈ On)
7 noxp1o 31063 . . . 4 ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} ∈ On → ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) ∈ No )
86, 7syl 17 . . 3 ((𝐴 No 𝐴 ∈ V) → ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) ∈ No )
98adantr 480 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) ∈ No )
10 ssel2 3563 . . . . . 6 ((𝐴 No 𝑦𝐴) → 𝑦 No )
1110adantlr 747 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → 𝑦 No )
124nobndlem4 31094 . . . . . . . 8 (𝑦 No {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ On)
1310, 12syl 17 . . . . . . 7 ((𝐴 No 𝑦𝐴) → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ On)
1413adantlr 747 . . . . . 6 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ On)
156adantr 480 . . . . . . . . . . 11 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} ∈ On)
164, 5nobndlem6 31096 . . . . . . . . . . . 12 ((𝐴 No 𝑦𝐴) → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜})
1716adantlr 747 . . . . . . . . . . 11 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜})
18 onelss 5683 . . . . . . . . . . 11 ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} ∈ On → ( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜}))
1915, 17, 18sylc 63 . . . . . . . . . 10 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜})
2019sselda 3568 . . . . . . . . 9 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → 𝑑 {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜})
213fvconst2 6374 . . . . . . . . 9 (𝑑 {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = 1𝑜)
2220, 21syl 17 . . . . . . . 8 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = 1𝑜)
2314adantr 480 . . . . . . . . . . . . . . . 16 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ On)
24 onss 6882 . . . . . . . . . . . . . . . . . 18 ( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ On → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ On)
2514, 24syl 17 . . . . . . . . . . . . . . . . 17 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ On)
2625sselda 3568 . . . . . . . . . . . . . . . 16 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → 𝑑 ∈ On)
27 ontri1 5674 . . . . . . . . . . . . . . . 16 (( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ On ∧ 𝑑 ∈ On) → ( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑 ↔ ¬ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
2823, 26, 27syl2anc 691 . . . . . . . . . . . . . . 15 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → ( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑 ↔ ¬ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
2928biimpd 218 . . . . . . . . . . . . . 14 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → ( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑 → ¬ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
3029con2d 128 . . . . . . . . . . . . 13 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → (𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → ¬ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑))
3130ex 449 . . . . . . . . . . . 12 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → (𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → (𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → ¬ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑)))
3231pm2.43d 51 . . . . . . . . . . 11 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → (𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → ¬ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑))
3332imp 444 . . . . . . . . . 10 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → ¬ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑)
34 intss1 4427 . . . . . . . . . 10 (𝑑 ∈ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ⊆ 𝑑)
3533, 34nsyl 134 . . . . . . . . 9 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → ¬ 𝑑 ∈ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜})
36 df-ne 2782 . . . . . . . . . 10 ((𝑦𝑑) ≠ 1𝑜 ↔ ¬ (𝑦𝑑) = 1𝑜)
37 fveq2 6103 . . . . . . . . . . . . . 14 (𝑘 = 𝑑 → (𝑦𝑘) = (𝑦𝑑))
3837neeq1d 2841 . . . . . . . . . . . . 13 (𝑘 = 𝑑 → ((𝑦𝑘) ≠ 1𝑜 ↔ (𝑦𝑑) ≠ 1𝑜))
3938elrab3 3332 . . . . . . . . . . . 12 (𝑑 ∈ On → (𝑑 ∈ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ↔ (𝑦𝑑) ≠ 1𝑜))
4039biimprd 237 . . . . . . . . . . 11 (𝑑 ∈ On → ((𝑦𝑑) ≠ 1𝑜𝑑 ∈ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
4126, 40syl 17 . . . . . . . . . 10 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → ((𝑦𝑑) ≠ 1𝑜𝑑 ∈ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
4236, 41syl5bir 232 . . . . . . . . 9 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → (¬ (𝑦𝑑) = 1𝑜𝑑 ∈ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
4335, 42mt3d 139 . . . . . . . 8 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → (𝑦𝑑) = 1𝑜)
4422, 43eqtr4d 2647 . . . . . . 7 ((((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) ∧ 𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑))
4544ralrimiva 2949 . . . . . 6 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ∀𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑))
463fvconst2 6374 . . . . . . . 8 ( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 1𝑜)
4717, 46syl 17 . . . . . . 7 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 1𝑜)
484nobndlem5 31095 . . . . . . . . . . . . . 14 (𝑦 No → (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) ≠ 1𝑜)
4911, 48syl 17 . . . . . . . . . . . . 13 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) ≠ 1𝑜)
5049neneqd 2787 . . . . . . . . . . . 12 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ¬ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 1𝑜)
51 nofv 31054 . . . . . . . . . . . . 13 (𝑦 No → ((𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅ ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 1𝑜 ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜))
5211, 51syl 17 . . . . . . . . . . . 12 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ((𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅ ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 1𝑜 ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜))
53 3orel2 30847 . . . . . . . . . . . 12 (¬ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 1𝑜 → (((𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅ ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 1𝑜 ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜) → ((𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅ ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
5450, 52, 53sylc 63 . . . . . . . . . . 11 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ((𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅ ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜))
55 eqid 2610 . . . . . . . . . . 11 1𝑜 = 1𝑜
5654, 55jctil 558 . . . . . . . . . 10 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → (1𝑜 = 1𝑜 ∧ ((𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅ ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
57 andi 907 . . . . . . . . . 10 ((1𝑜 = 1𝑜 ∧ ((𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅ ∨ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)) ↔ ((1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅) ∨ (1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
5856, 57sylib 207 . . . . . . . . 9 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ((1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅) ∨ (1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
5958orcd 406 . . . . . . . 8 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → (((1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅) ∨ (1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)) ∨ (1𝑜 = ∅ ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
60 fvex 6113 . . . . . . . . . 10 (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) ∈ V
613, 60brtp 30892 . . . . . . . . 9 (1𝑜{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) ↔ ((1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅) ∨ (1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜) ∨ (1𝑜 = ∅ ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
62 df-3or 1032 . . . . . . . . 9 (((1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅) ∨ (1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜) ∨ (1𝑜 = ∅ ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)) ↔ (((1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅) ∨ (1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)) ∨ (1𝑜 = ∅ ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
6361, 62bitri 263 . . . . . . . 8 (1𝑜{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) ↔ (((1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = ∅) ∨ (1𝑜 = 1𝑜 ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)) ∨ (1𝑜 = ∅ ∧ (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}) = 2𝑜)))
6459, 63sylibr 223 . . . . . . 7 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → 1𝑜{⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
6547, 64eqbrtrd 4605 . . . . . 6 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
66 raleq 3115 . . . . . . . 8 (𝑐 = {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → (∀𝑑𝑐 (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ↔ ∀𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑)))
67 fveq2 6103 . . . . . . . . 9 (𝑐 = {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑐) = (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
68 fveq2 6103 . . . . . . . . 9 (𝑐 = {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → (𝑦𝑐) = (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))
6967, 68breq12d 4596 . . . . . . . 8 (𝑐 = {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → ((( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑐){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦𝑐) ↔ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜})))
7066, 69anbi12d 743 . . . . . . 7 (𝑐 = {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} → ((∀𝑑𝑐 (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ∧ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑐){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦𝑐)) ↔ (∀𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ∧ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))))
7170rspcev 3282 . . . . . 6 (( {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} ∈ On ∧ (∀𝑑 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜} (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ∧ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘ {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦 {𝑘 ∈ On ∣ (𝑦𝑘) ≠ 1𝑜}))) → ∃𝑐 ∈ On (∀𝑑𝑐 (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ∧ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑐){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦𝑐)))
7214, 45, 65, 71syl12anc 1316 . . . . 5 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ∃𝑐 ∈ On (∀𝑑𝑐 (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ∧ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑐){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦𝑐)))
73 sltval 31044 . . . . . 6 ((( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) ∈ No 𝑦 No ) → (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦 ↔ ∃𝑐 ∈ On (∀𝑑𝑐 (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ∧ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑐){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦𝑐))))
7473biimpar 501 . . . . 5 (((( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) ∈ No 𝑦 No ) ∧ ∃𝑐 ∈ On (∀𝑑𝑐 (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑑) = (𝑦𝑑) ∧ (( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})‘𝑐){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝑦𝑐))) → ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦)
759, 11, 72, 74syl21anc 1317 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ 𝑦𝐴) → ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦)
7675ralrimiva 2949 . . 3 ((𝐴 No 𝐴 ∈ V) → ∀𝑦𝐴 ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦)
774, 5nobndlem8 31098 . . 3 ((𝐴 No 𝐴 ∈ V) → ( bday ‘( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})) ⊆ suc ( bday 𝐴))
78 breq1 4586 . . . . . 6 (𝑥 = ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) → (𝑥 <s 𝑦 ↔ ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦))
7978ralbidv 2969 . . . . 5 (𝑥 = ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) → (∀𝑦𝐴 𝑥 <s 𝑦 ↔ ∀𝑦𝐴 ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦))
80 fveq2 6103 . . . . . 6 (𝑥 = ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) → ( bday 𝑥) = ( bday ‘( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})))
8180sseq1d 3595 . . . . 5 (𝑥 = ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) → (( bday 𝑥) ⊆ suc ( bday 𝐴) ↔ ( bday ‘( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})) ⊆ suc ( bday 𝐴)))
8279, 81anbi12d 743 . . . 4 (𝑥 = ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) → ((∀𝑦𝐴 𝑥 <s 𝑦 ∧ ( bday 𝑥) ⊆ suc ( bday 𝐴)) ↔ (∀𝑦𝐴 ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦 ∧ ( bday ‘( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})) ⊆ suc ( bday 𝐴))))
8382rspcev 3282 . . 3 ((( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) ∈ No ∧ (∀𝑦𝐴 ( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜}) <s 𝑦 ∧ ( bday ‘( {𝑎 ∈ On ∣ ∀𝑛𝐴𝑏𝑎 (𝑛𝑏) ≠ 1𝑜} × {1𝑜})) ⊆ suc ( bday 𝐴))) → ∃𝑥 No (∀𝑦𝐴 𝑥 <s 𝑦 ∧ ( bday 𝑥) ⊆ suc ( bday 𝐴)))
848, 76, 77, 83syl12anc 1316 . 2 ((𝐴 No 𝐴 ∈ V) → ∃𝑥 No (∀𝑦𝐴 𝑥 <s 𝑦 ∧ ( bday 𝑥) ⊆ suc ( bday 𝐴)))
851, 84sylan2 490 1 ((𝐴 No 𝐴𝑉) → ∃𝑥 No (∀𝑦𝐴 𝑥 <s 𝑦 ∧ ( bday 𝑥) ⊆ suc ( bday 𝐴)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  {csn 4125  {ctp 4129  ⟨cop 4131  ∪ cuni 4372  ∩ cint 4410   class class class wbr 4583   × cxp 5036   “ cima 5041  Oncon0 5640  suc csuc 5642  ‘cfv 5804  1𝑜c1o 7440  2𝑜c2o 7441   No csur 31037
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