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Theorem ackbij1lem18 9243
Description: Lemma for ackbij1 9244. (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 3872 . . . 4 (𝐴 (ω ∖ 𝐴)) ⊆ 𝐴
2 ackbij.f . . . . 5 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
32ackbij1lem11 9236 . . . 4 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 (ω ∖ 𝐴)) ⊆ 𝐴) → (𝐴 (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin))
41, 3mpan2 709 . . 3 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐴 (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin))
5 difss 3872 . . . . . . 7 (ω ∖ 𝐴) ⊆ ω
6 omsson 7226 . . . . . . 7 ω ⊆ On
75, 6sstri 3745 . . . . . 6 (ω ∖ 𝐴) ⊆ On
8 ominf 8329 . . . . . . . 8 ¬ ω ∈ Fin
9 inss2 3969 . . . . . . . . 9 (𝒫 ω ∩ Fin) ⊆ Fin
109sseli 3732 . . . . . . . 8 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
11 difinf 8387 . . . . . . . 8 ((¬ ω ∈ Fin ∧ 𝐴 ∈ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin)
128, 10, 11sylancr 698 . . . . . . 7 (𝐴 ∈ (𝒫 ω ∩ Fin) → ¬ (ω ∖ 𝐴) ∈ Fin)
13 0fin 8345 . . . . . . . . 9 ∅ ∈ Fin
14 eleq1 2819 . . . . . . . . 9 ((ω ∖ 𝐴) = ∅ → ((ω ∖ 𝐴) ∈ Fin ↔ ∅ ∈ Fin))
1513, 14mpbiri 248 . . . . . . . 8 ((ω ∖ 𝐴) = ∅ → (ω ∖ 𝐴) ∈ Fin)
1615necon3bi 2950 . . . . . . 7 (¬ (ω ∖ 𝐴) ∈ Fin → (ω ∖ 𝐴) ≠ ∅)
1712, 16syl 17 . . . . . 6 (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ≠ ∅)
18 onint 7152 . . . . . 6 (((ω ∖ 𝐴) ⊆ On ∧ (ω ∖ 𝐴) ≠ ∅) → (ω ∖ 𝐴) ∈ (ω ∖ 𝐴))
197, 17, 18sylancr 698 . . . . 5 (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ∈ (ω ∖ 𝐴))
2019eldifad 3719 . . . 4 (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ∈ ω)
21 ackbij1lem4 9229 . . . 4 ( (ω ∖ 𝐴) ∈ ω → { (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin))
2220, 21syl 17 . . 3 (𝐴 ∈ (𝒫 ω ∩ Fin) → { (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin))
23 ackbij1lem6 9231 . . 3 (((𝐴 (ω ∖ 𝐴)) ∈ (𝒫 ω ∩ Fin) ∧ { (ω ∖ 𝐴)} ∈ (𝒫 ω ∩ Fin)) → ((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)}) ∈ (𝒫 ω ∩ Fin))
244, 22, 23syl2anc 696 . 2 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)}) ∈ (𝒫 ω ∩ Fin))
2519eldifbd 3720 . . . . . 6 (𝐴 ∈ (𝒫 ω ∩ Fin) → ¬ (ω ∖ 𝐴) ∈ 𝐴)
26 disjsn 4382 . . . . . 6 ((𝐴 ∩ { (ω ∖ 𝐴)}) = ∅ ↔ ¬ (ω ∖ 𝐴) ∈ 𝐴)
2725, 26sylibr 224 . . . . 5 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐴 ∩ { (ω ∖ 𝐴)}) = ∅)
28 ssdisj 4162 . . . . 5 (((𝐴 (ω ∖ 𝐴)) ⊆ 𝐴 ∧ (𝐴 ∩ { (ω ∖ 𝐴)}) = ∅) → ((𝐴 (ω ∖ 𝐴)) ∩ { (ω ∖ 𝐴)}) = ∅)
291, 27, 28sylancr 698 . . . 4 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 (ω ∖ 𝐴)) ∩ { (ω ∖ 𝐴)}) = ∅)
302ackbij1lem9 9234 . . . 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 1473 . . 3 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹‘{ (ω ∖ 𝐴)})))
322ackbij1lem14 9239 . . . . 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 6821 . . 3 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹‘{ (ω ∖ 𝐴)})) = ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 suc (𝐹 (ω ∖ 𝐴))))
352ackbij1lem10 9235 . . . . . . 7 𝐹:(𝒫 ω ∩ Fin)⟶ω
3635ffvelrni 6513 . . . . . 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 9228 . . . . . . 7 ( (ω ∖ 𝐴) ∈ ω → (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin))
3920, 38syl 17 . . . . . 6 (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin))
4035ffvelrni 6513 . . . . . 6 ( (ω ∖ 𝐴) ∈ (𝒫 ω ∩ Fin) → (𝐹 (ω ∖ 𝐴)) ∈ ω)
4139, 40syl 17 . . . . 5 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹 (ω ∖ 𝐴)) ∈ ω)
42 nnasuc 7847 . . . . 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 696 . . . 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 3940 . . . . . . . . 9 ((𝐴 (ω ∖ 𝐴)) ∩ (ω ∖ 𝐴)) = ( (ω ∖ 𝐴) ∩ (𝐴 (ω ∖ 𝐴)))
45 disjdif 4176 . . . . . . . . 9 ( (ω ∖ 𝐴) ∩ (𝐴 (ω ∖ 𝐴))) = ∅
4644, 45eqtri 2774 . . . . . . . 8 ((𝐴 (ω ∖ 𝐴)) ∩ (ω ∖ 𝐴)) = ∅
4746a1i 11 . . . . . . 7 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 (ω ∖ 𝐴)) ∩ (ω ∖ 𝐴)) = ∅)
482ackbij1lem9 9234 . . . . . . 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 1473 . . . . . 6 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ (ω ∖ 𝐴))) = ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹 (ω ∖ 𝐴))))
50 uncom 3892 . . . . . . . 8 ((𝐴 (ω ∖ 𝐴)) ∪ (ω ∖ 𝐴)) = ( (ω ∖ 𝐴) ∪ (𝐴 (ω ∖ 𝐴)))
51 onnmin 7160 . . . . . . . . . . . . . . 15 (((ω ∖ 𝐴) ⊆ On ∧ 𝑎 ∈ (ω ∖ 𝐴)) → ¬ 𝑎 (ω ∖ 𝐴))
527, 51mpan 708 . . . . . . . . . . . . . 14 (𝑎 ∈ (ω ∖ 𝐴) → ¬ 𝑎 (ω ∖ 𝐴))
5352con2i 134 . . . . . . . . . . . . 13 (𝑎 (ω ∖ 𝐴) → ¬ 𝑎 ∈ (ω ∖ 𝐴))
5453adantl 473 . . . . . . . . . . . 12 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 (ω ∖ 𝐴)) → ¬ 𝑎 ∈ (ω ∖ 𝐴))
55 ordom 7231 . . . . . . . . . . . . . . 15 Ord ω
56 ordelss 5892 . . . . . . . . . . . . . . 15 ((Ord ω ∧ (ω ∖ 𝐴) ∈ ω) → (ω ∖ 𝐴) ⊆ ω)
5755, 20, 56sylancr 698 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ⊆ ω)
5857sselda 3736 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 (ω ∖ 𝐴)) → 𝑎 ∈ ω)
59 eldif 3717 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (ω ∖ 𝐴) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎𝐴))
6059simplbi2 656 . . . . . . . . . . . . . . 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 65 . . . . . . . . . . 11 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 (ω ∖ 𝐴)) → 𝑎𝐴)
6665ex 449 . . . . . . . . . 10 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝑎 (ω ∖ 𝐴) → 𝑎𝐴))
6766ssrdv 3742 . . . . . . . . 9 (𝐴 ∈ (𝒫 ω ∩ Fin) → (ω ∖ 𝐴) ⊆ 𝐴)
68 undif 4185 . . . . . . . . 9 ( (ω ∖ 𝐴) ⊆ 𝐴 ↔ ( (ω ∖ 𝐴) ∪ (𝐴 (ω ∖ 𝐴))) = 𝐴)
6967, 68sylib 208 . . . . . . . 8 (𝐴 ∈ (𝒫 ω ∩ Fin) → ( (ω ∖ 𝐴) ∪ (𝐴 (ω ∖ 𝐴))) = 𝐴)
7050, 69syl5eq 2798 . . . . . . 7 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐴 (ω ∖ 𝐴)) ∪ (ω ∖ 𝐴)) = 𝐴)
7170fveq2d 6348 . . . . . 6 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ (ω ∖ 𝐴))) = (𝐹𝐴))
7249, 71eqtr3d 2788 . . . . 5 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 (𝐹 (ω ∖ 𝐴))) = (𝐹𝐴))
73 suceq 5943 . . . . 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 2786 . . 3 (𝐴 ∈ (𝒫 ω ∩ Fin) → ((𝐹‘(𝐴 (ω ∖ 𝐴))) +𝑜 suc (𝐹 (ω ∖ 𝐴))) = suc (𝐹𝐴))
7631, 34, 753eqtrd 2790 . 2 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)})) = suc (𝐹𝐴))
77 fveq2 6344 . . . 4 (𝑏 = ((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)}) → (𝐹𝑏) = (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)})))
7877eqeq1d 2754 . . 3 (𝑏 = ((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)}) → ((𝐹𝑏) = suc (𝐹𝐴) ↔ (𝐹‘((𝐴 (ω ∖ 𝐴)) ∪ { (ω ∖ 𝐴)})) = suc (𝐹𝐴)))
7978rspcev 3441 . 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 696 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 1624  wcel 2131  wne 2924  wrex 3043  cdif 3704  cun 3705  cin 3706  wss 3707  c0 4050  𝒫 cpw 4294  {csn 4313   cint 4619   ciun 4664  cmpt 4873   × cxp 5256  Ord word 5875  Oncon0 5876  suc csuc 5878  cfv 6041  (class class class)co 6805  ωcom 7222   +𝑜 coa 7718  Fincfn 8113  cardccrd 8943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8947  df-cda 9174
This theorem is referenced by:  ackbij1  9244
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