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Theorem ackbij1lem18 8942
 Description: Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Hypothesis
Ref Expression
ackbij.f 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
Assertion
Ref Expression
ackbij1lem18 (𝐴 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc (𝐹𝐴))
Distinct variable groups:   𝐹,𝑏,𝑥,𝑦   𝐴,𝑏,𝑥,𝑦

Proof of Theorem ackbij1lem18
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 difss 3699 . . . 4 (𝐴 (ω ∖ 𝐴)) ⊆ 𝐴
2 ackbij.f . . . . 5 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
32ackbij1lem11 8935 . . . 4 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 (ω ∖ 𝐴)) ⊆ 𝐴) → (𝐴 (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin))
41, 3mpan2 703 . . 3 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐴 (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin))
5 difss 3699 . . . . . . 7 (ω ∖ 𝐴) ⊆ ω
6 omsson 6961 . . . . . . 7 ω ⊆ On
75, 6sstri 3577 . . . . . 6 (ω ∖ 𝐴) ⊆ On
8 ominf 8057 . . . . . . . 8 ¬ ω ∈ Fin
9 inss2 3796 . . . . . . . . 9 (𝒫 ω ∩ Fin) ⊆ Fin
109sseli 3564 . . . . . . . 8 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
11 difinf 8115 . . . . . . . 8 ((¬ ω ∈ Fin ∧ 𝐴 ∈ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin)
128, 10, 11sylancr 694 . . . . . . 7 (𝐴 ∈ (𝒫 ω ∩ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin)
13 0fin 8073 . . . . . . . . 9 ∅ ∈ Fin
14 eleq1 2676 . . . . . . . . 9 ((ω ∖ 𝐴) = ∅ → ((ω ∖ 𝐴) ∈ Fin ↔ ∅ ∈ Fin))
1513, 14mpbiri 247 . . . . . . . 8 ((ω ∖ 𝐴) = ∅ → (ω ∖ 𝐴) ∈ Fin)
1615necon3bi 2808 . . . . . . 7 (¬ (ω ∖ 𝐴) ∈ Fin → (ω ∖ 𝐴) ≠ ∅)
1712, 16syl 17 . . . . . 6 (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ≠ ∅)
18 onint 6887 . . . . . 6 (((ω ∖ 𝐴) ⊆ On ∧ (ω ∖ 𝐴) ≠ ∅) → (ω ∖ 𝐴) ∈ (ω ∖ 𝐴))
197, 17, 18sylancr 694 . . . . 5 (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ∈ (ω ∖ 𝐴))
2019eldifad 3552 . . . 4 (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ∈ ω)
21 ackbij1lem4 8928 . . . 4 ( (ω ∖ 𝐴) ∈ ω → { (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin))
2220, 21syl 17 . . 3 (𝐴 ∈ (𝒫 ω ∩ Fin) → { (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin))
23 ackbij1lem6 8930 . . 3 (((𝐴 (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin) ∧ { (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin)) → ((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)}) ∈ (𝒫 ω ∩ Fin))
244, 22, 23syl2anc 691 . 2 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)}) ∈ (𝒫 ω ∩ Fin))
2519eldifbd 3553 . . . . . 6 (𝐴 ∈ (𝒫 ω ∩ Fin) → ¬ (ω ∖ 𝐴) ∈ 𝐴)
26 disjsn 4192 . . . . . 6 ((𝐴 ∩ { (ω ∖ 𝐴)}) = ∅ ↔ ¬ (ω ∖ 𝐴) ∈ 𝐴)
2725, 26sylibr 223 . . . . 5 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐴 ∩ { (ω ∖ 𝐴)}) = ∅)
28 ssdisj 3978 . . . . 5 (((𝐴 (ω ∖ 𝐴)) ⊆ 𝐴 ∧ (𝐴 ∩ { (ω ∖ 𝐴)}) = ∅) → ((𝐴 (ω ∖ 𝐴)) ∩ { (ω ∖ 𝐴)}) = ∅)
291, 27, 28sylancr 694 . . . 4 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 (ω ∖ 𝐴)) ∩ { (ω ∖ 𝐴)}) = ∅)
302ackbij1lem9 8933 . . . 4 (((𝐴 (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin) ∧ { (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin) ∧ ((𝐴 (ω ∖ 𝐴)) ∩ { (ω ∖ 𝐴)}) = ∅) → (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹‘{ (ω ∖ 𝐴)})))
314, 22, 29, 30syl3anc 1318 . . 3 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹‘{ (ω ∖ 𝐴)})))
322ackbij1lem14 8938 . . . . 5 ( (ω ∖ 𝐴) ∈ ω → (𝐹‘{ (ω ∖ 𝐴)}) = suc (𝐹 (ω ∖ 𝐴)))
3320, 32syl 17 . . . 4 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘{ (ω ∖ 𝐴)}) = suc (𝐹 (ω ∖ 𝐴)))
3433oveq2d 6565 . . 3 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹‘{ (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 suc (𝐹 (ω ∖ 𝐴))))
352ackbij1lem10 8934 . . . . . . 7 𝐹:(𝒫 ω ∩ Fin)⟶ω
3635ffvelrni 6266 . . . . . 6 ((𝐴 (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐴 (ω ∖ 𝐴))) ∈ ω)
374, 36syl 17 . . . . 5 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐴 (ω ∖ 𝐴))) ∈ ω)
38 ackbij1lem3 8927 . . . . . . 7 ( (ω ∖ 𝐴) ∈ ω → (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin))
3920, 38syl 17 . . . . . 6 (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin))
4035ffvelrni 6266 . . . . . 6 ( (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin) → (𝐹 (ω ∖ 𝐴)) ∈ ω)
4139, 40syl 17 . . . . 5 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹 (ω ∖ 𝐴)) ∈ ω)
42 nnasuc 7573 . . . . 5 (((𝐹‘(𝐴 (ω ∖ 𝐴))) ∈ ω ∧ (𝐹 (ω ∖ 𝐴)) ∈ ω) → ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 suc (𝐹 (ω ∖ 𝐴))) = suc ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹 (ω ∖ 𝐴))))
4337, 41, 42syl2anc 691 . . . 4 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 suc (𝐹 (ω ∖ 𝐴))) = suc ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹 (ω ∖ 𝐴))))
44 incom 3767 . . . . . . . . 9 ((𝐴 (ω ∖ 𝐴)) ∩ (ω ∖ 𝐴)) = ( (ω ∖ 𝐴) ∩ (𝐴 (ω ∖ 𝐴)))
45 disjdif 3992 . . . . . . . . 9 ( (ω ∖ 𝐴) ∩ (𝐴 (ω ∖ 𝐴))) = ∅
4644, 45eqtri 2632 . . . . . . . 8 ((𝐴 (ω ∖ 𝐴)) ∩ (ω ∖ 𝐴)) = ∅
4746a1i 11 . . . . . . 7 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 (ω ∖ 𝐴)) ∩ (ω ∖ 𝐴)) = ∅)
482ackbij1lem9 8933 . . . . . . 7 (((𝐴 (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin) ∧ (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin) ∧ ((𝐴 (ω ∖ 𝐴)) ∩ (ω ∖ 𝐴)) = ∅) → (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ (ω ∖ 𝐴))) = ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹 (ω ∖ 𝐴))))
494, 39, 47, 48syl3anc 1318 . . . . . 6 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ (ω ∖ 𝐴))) = ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹 (ω ∖ 𝐴))))
50 uncom 3719 . . . . . . . 8 ((𝐴 (ω ∖ 𝐴)) ∪ (ω ∖ 𝐴)) = ( (ω ∖ 𝐴) ∪ (𝐴 (ω ∖ 𝐴)))
51 onnmin 6895 . . . . . . . . . . . . . . 15 (((ω ∖ 𝐴) ⊆ On ∧ 𝑎 ∈ (ω ∖ 𝐴)) → ¬ 𝑎 (ω ∖ 𝐴))
527, 51mpan 702 . . . . . . . . . . . . . 14 (𝑎 ∈ (ω ∖ 𝐴) → ¬ 𝑎 (ω ∖ 𝐴))
5352con2i 133 . . . . . . . . . . . . 13 (𝑎 (ω ∖ 𝐴) → ¬ 𝑎 ∈ (ω ∖ 𝐴))
5453adantl 481 . . . . . . . . . . . 12 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 (ω ∖ 𝐴)) → ¬ 𝑎 ∈ (ω ∖ 𝐴))
55 ordom 6966 . . . . . . . . . . . . . . 15 Ord ω
56 ordelss 5656 . . . . . . . . . . . . . . 15 ((Ord ω ∧ (ω ∖ 𝐴) ∈ ω) → (ω ∖ 𝐴) ⊆ ω)
5755, 20, 56sylancr 694 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ⊆ ω)
5857sselda 3568 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 (ω ∖ 𝐴)) → 𝑎 ∈ ω)
59 eldif 3550 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (ω ∖ 𝐴) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎𝐴))
6059simplbi2 653 . . . . . . . . . . . . . . 15 (𝑎 ∈ ω → (¬ 𝑎𝐴𝑎 ∈ (ω ∖ 𝐴)))
6160orrd 392 . . . . . . . . . . . . . 14 (𝑎 ∈ ω → (𝑎𝐴𝑎 ∈ (ω ∖ 𝐴)))
6261orcomd 402 . . . . . . . . . . . . 13 (𝑎 ∈ ω → (𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎𝐴))
6358, 62syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 (ω ∖ 𝐴)) → (𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎𝐴))
64 orel1 396 . . . . . . . . . . . 12 𝑎 ∈ (ω ∖ 𝐴) → ((𝑎 ∈ (ω ∖ 𝐴) ∨ 𝑎𝐴) → 𝑎𝐴))
6554, 63, 64sylc 63 . . . . . . . . . . 11 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 (ω ∖ 𝐴)) → 𝑎𝐴)
6665ex 449 . . . . . . . . . 10 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝑎 (ω ∖ 𝐴) → 𝑎𝐴))
6766ssrdv 3574 . . . . . . . . 9 (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ⊆ 𝐴)
68 undif 4001 . . . . . . . . 9 ( (ω ∖ 𝐴) ⊆ 𝐴 ↔ ( (ω ∖ 𝐴) ∪ (𝐴 (ω ∖ 𝐴))) = 𝐴)
6967, 68sylib 207 . . . . . . . 8 (𝐴 ∈ (𝒫 ω ∩ Fin) → ( (ω ∖ 𝐴) ∪ (𝐴 (ω ∖ 𝐴))) = 𝐴)
7050, 69syl5eq 2656 . . . . . . 7 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 (ω ∖ 𝐴)) ∪ (ω ∖ 𝐴)) = 𝐴)
7170fveq2d 6107 . . . . . 6 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ (ω ∖ 𝐴))) = (𝐹𝐴))
7249, 71eqtr3d 2646 . . . . 5 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹 (ω ∖ 𝐴))) = (𝐹𝐴))
73 suceq 5707 . . . . 5 (((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹 (ω ∖ 𝐴))) = (𝐹𝐴) → suc ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹 (ω ∖ 𝐴))) = suc (𝐹𝐴))
7472, 73syl 17 . . . 4 (𝐴 ∈ (𝒫 ω ∩ Fin) → suc ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹 (ω ∖ 𝐴))) = suc (𝐹𝐴))
7543, 74eqtrd 2644 . . 3 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 suc (𝐹 (ω ∖ 𝐴))) = suc (𝐹𝐴))
7631, 34, 753eqtrd 2648 . 2 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)})) = suc (𝐹𝐴))
77 fveq2 6103 . . . 4 (𝑏 = ((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)}) → (𝐹𝑏) = (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)})))
7877eqeq1d 2612 . . 3 (𝑏 = ((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)}) → ((𝐹𝑏) = suc (𝐹𝐴) ↔ (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)})) = suc (𝐹𝐴)))
7978rspcev 3282 . 2 ((((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)}) ∈ (𝒫 ω ∩ Fin) ∧ (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)})) = suc (𝐹𝐴)) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc (𝐹𝐴))
8024, 76, 79syl2anc 691 1 (𝐴 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹𝑏) = suc (𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∩ cint 4410  ∪ ciun 4455   ↦ cmpt 4643   × cxp 5036  Ord word 5639  Oncon0 5640  suc csuc 5642  ‘cfv 5804  (class class class)co 6549  ωcom 6957   +𝑜 coa 7444  Fincfn 7841  cardccrd 8644 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 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-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-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-card 8648  df-cda 8873 This theorem is referenced by:  ackbij1  8943
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