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Theorem ssnnfiOLD 9193
Description: Obsolete version of ssnnfi 9192 as of 23-Sep-2024. (Contributed by NM, 24-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ssnnfiOLD ((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐵 ∈ Fin)

Proof of Theorem ssnnfiOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sspss 4091 . . 3 (𝐵𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))
2 pssnn 9191 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ∃𝑥𝐴 𝐵𝑥)
3 elnn 7879 . . . . . . . . 9 ((𝑥𝐴𝐴 ∈ ω) → 𝑥 ∈ ω)
43expcom 412 . . . . . . . 8 (𝐴 ∈ ω → (𝑥𝐴𝑥 ∈ ω))
54anim1d 609 . . . . . . 7 (𝐴 ∈ ω → ((𝑥𝐴𝐵𝑥) → (𝑥 ∈ ω ∧ 𝐵𝑥)))
65reximdv2 3154 . . . . . 6 (𝐴 ∈ ω → (∃𝑥𝐴 𝐵𝑥 → ∃𝑥 ∈ ω 𝐵𝑥))
76adantr 479 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵𝐴) → (∃𝑥𝐴 𝐵𝑥 → ∃𝑥 ∈ ω 𝐵𝑥))
82, 7mpd 15 . . . 4 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ∃𝑥 ∈ ω 𝐵𝑥)
9 eleq1 2813 . . . . . 6 (𝐵 = 𝐴 → (𝐵 ∈ ω ↔ 𝐴 ∈ ω))
109biimparc 478 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐵 ∈ ω)
11 enrefnn 9070 . . . . 5 (𝐵 ∈ ω → 𝐵𝐵)
12 breq2 5147 . . . . . 6 (𝑥 = 𝐵 → (𝐵𝑥𝐵𝐵))
1312rspcev 3601 . . . . 5 ((𝐵 ∈ ω ∧ 𝐵𝐵) → ∃𝑥 ∈ ω 𝐵𝑥)
1410, 11, 13syl2anc2 583 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → ∃𝑥 ∈ ω 𝐵𝑥)
158, 14jaodan 955 . . 3 ((𝐴 ∈ ω ∧ (𝐵𝐴𝐵 = 𝐴)) → ∃𝑥 ∈ ω 𝐵𝑥)
161, 15sylan2b 592 . 2 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ∃𝑥 ∈ ω 𝐵𝑥)
17 isfi 8995 . 2 (𝐵 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐵𝑥)
1816, 17sylibr 233 1 ((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐵 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845   = wceq 1533  wcel 2098  wrex 3060  wss 3939  wpss 3940   class class class wbr 5143  ωcom 7868  cen 8959  Fincfn 8962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-om 7869  df-en 8963  df-fin 8966
This theorem is referenced by: (None)
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