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Theorem fineqvlem 8118
Description: Lemma for fineqv 8119. (Contributed by Mario Carneiro, 20-Jan-2013.) (Proof shortened by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
fineqvlem ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝒫 𝒫 𝐴)

Proof of Theorem fineqvlem
Dummy variables 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4810 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
21adantr 481 . . 3 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → 𝒫 𝐴 ∈ V)
3 pwexg 4810 . . 3 (𝒫 𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V)
42, 3syl 17 . 2 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → 𝒫 𝒫 𝐴 ∈ V)
5 ssrab2 3666 . . . . 5 {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ⊆ 𝒫 𝐴
6 elpw2g 4787 . . . . . 6 (𝒫 𝐴 ∈ V → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} ∈ 𝒫 𝒫 𝐴 ↔ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ⊆ 𝒫 𝐴))
72, 6syl 17 . . . . 5 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} ∈ 𝒫 𝒫 𝐴 ↔ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ⊆ 𝒫 𝐴))
85, 7mpbiri 248 . . . 4 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ∈ 𝒫 𝒫 𝐴)
98a1d 25 . . 3 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → (𝑏 ∈ ω → {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ∈ 𝒫 𝒫 𝐴))
10 isinf 8117 . . . . . . . . 9 𝐴 ∈ Fin → ∀𝑏 ∈ ω ∃𝑒(𝑒𝐴𝑒𝑏))
1110r19.21bi 2927 . . . . . . . 8 ((¬ 𝐴 ∈ Fin ∧ 𝑏 ∈ ω) → ∃𝑒(𝑒𝐴𝑒𝑏))
1211ad2ant2lr 783 . . . . . . 7 (((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ∃𝑒(𝑒𝐴𝑒𝑏))
13 selpw 4137 . . . . . . . . . . 11 (𝑒 ∈ 𝒫 𝐴𝑒𝐴)
1413biimpri 218 . . . . . . . . . 10 (𝑒𝐴𝑒 ∈ 𝒫 𝐴)
1514anim1i 591 . . . . . . . . 9 ((𝑒𝐴𝑒𝑏) → (𝑒 ∈ 𝒫 𝐴𝑒𝑏))
16 breq1 4616 . . . . . . . . . 10 (𝑑 = 𝑒 → (𝑑𝑏𝑒𝑏))
1716elrab 3346 . . . . . . . . 9 (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ↔ (𝑒 ∈ 𝒫 𝐴𝑒𝑏))
1815, 17sylibr 224 . . . . . . . 8 ((𝑒𝐴𝑒𝑏) → 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏})
1918eximi 1759 . . . . . . 7 (∃𝑒(𝑒𝐴𝑒𝑏) → ∃𝑒 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏})
2012, 19syl 17 . . . . . 6 (((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ∃𝑒 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏})
21 eleq2 2687 . . . . . . . . 9 ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ↔ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐}))
2221biimpcd 239 . . . . . . . 8 (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐}))
2322adantl 482 . . . . . . 7 ((((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏}) → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐}))
2417simprbi 480 . . . . . . . . . 10 (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} → 𝑒𝑏)
25 breq1 4616 . . . . . . . . . . . 12 (𝑑 = 𝑒 → (𝑑𝑐𝑒𝑐))
2625elrab 3346 . . . . . . . . . . 11 (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐} ↔ (𝑒 ∈ 𝒫 𝐴𝑒𝑐))
2726simprbi 480 . . . . . . . . . 10 (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → 𝑒𝑐)
28 ensym 7949 . . . . . . . . . . 11 (𝑒𝑏𝑏𝑒)
29 entr 7952 . . . . . . . . . . 11 ((𝑏𝑒𝑒𝑐) → 𝑏𝑐)
3028, 29sylan 488 . . . . . . . . . 10 ((𝑒𝑏𝑒𝑐) → 𝑏𝑐)
3124, 27, 30syl2an 494 . . . . . . . . 9 ((𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐}) → 𝑏𝑐)
3231ex 450 . . . . . . . 8 (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏} → (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → 𝑏𝑐))
3332adantl 482 . . . . . . 7 ((((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏}) → (𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → 𝑏𝑐))
34 nneneq 8087 . . . . . . . . 9 ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏𝑐𝑏 = 𝑐))
3534biimpd 219 . . . . . . . 8 ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏𝑐𝑏 = 𝑐))
3635ad2antlr 762 . . . . . . 7 ((((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏}) → (𝑏𝑐𝑏 = 𝑐))
3723, 33, 363syld 60 . . . . . 6 ((((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) ∧ 𝑒 ∈ {𝑑 ∈ 𝒫 𝐴𝑑𝑏}) → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → 𝑏 = 𝑐))
3820, 37exlimddv 1860 . . . . 5 (((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐} → 𝑏 = 𝑐))
39 breq2 4617 . . . . . 6 (𝑏 = 𝑐 → (𝑑𝑏𝑑𝑐))
4039rabbidv 3177 . . . . 5 (𝑏 = 𝑐 → {𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐})
4138, 40impbid1 215 . . . 4 (((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ (𝑏 ∈ ω ∧ 𝑐 ∈ ω)) → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐} ↔ 𝑏 = 𝑐))
4241ex 450 . . 3 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → ({𝑑 ∈ 𝒫 𝐴𝑑𝑏} = {𝑑 ∈ 𝒫 𝐴𝑑𝑐} ↔ 𝑏 = 𝑐)))
439, 42dom2d 7940 . 2 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → (𝒫 𝒫 𝐴 ∈ V → ω ≼ 𝒫 𝒫 𝐴))
444, 43mpd 15 1 ((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝒫 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130   class class class wbr 4613  ωcom 7012  cen 7896  cdom 7897  Fincfn 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-er 7687  df-en 7900  df-dom 7901  df-fin 7903
This theorem is referenced by:  fineqv  8119  isfin1-2  9151
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