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Theorem ssnnfiOLD 9117
Description: Obsolete version of ssnnfi 9116 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 4060 . . 3 (𝐵𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))
2 pssnn 9115 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ∃𝑥𝐴 𝐵𝑥)
3 elnn 7814 . . . . . . . . 9 ((𝑥𝐴𝐴 ∈ ω) → 𝑥 ∈ ω)
43expcom 415 . . . . . . . 8 (𝐴 ∈ ω → (𝑥𝐴𝑥 ∈ ω))
54anim1d 612 . . . . . . 7 (𝐴 ∈ ω → ((𝑥𝐴𝐵𝑥) → (𝑥 ∈ ω ∧ 𝐵𝑥)))
65reximdv2 3158 . . . . . 6 (𝐴 ∈ ω → (∃𝑥𝐴 𝐵𝑥 → ∃𝑥 ∈ ω 𝐵𝑥))
76adantr 482 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵𝐴) → (∃𝑥𝐴 𝐵𝑥 → ∃𝑥 ∈ ω 𝐵𝑥))
82, 7mpd 15 . . . 4 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ∃𝑥 ∈ ω 𝐵𝑥)
9 eleq1 2822 . . . . . 6 (𝐵 = 𝐴 → (𝐵 ∈ ω ↔ 𝐴 ∈ ω))
109biimparc 481 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐵 ∈ ω)
11 enrefnn 8994 . . . . 5 (𝐵 ∈ ω → 𝐵𝐵)
12 breq2 5110 . . . . . 6 (𝑥 = 𝐵 → (𝐵𝑥𝐵𝐵))
1312rspcev 3580 . . . . 5 ((𝐵 ∈ ω ∧ 𝐵𝐵) → ∃𝑥 ∈ ω 𝐵𝑥)
1410, 11, 13syl2anc2 586 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → ∃𝑥 ∈ ω 𝐵𝑥)
158, 14jaodan 957 . . 3 ((𝐴 ∈ ω ∧ (𝐵𝐴𝐵 = 𝐴)) → ∃𝑥 ∈ ω 𝐵𝑥)
161, 15sylan2b 595 . 2 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ∃𝑥 ∈ ω 𝐵𝑥)
17 isfi 8919 . 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 397  wo 846   = wceq 1542  wcel 2107  wrex 3070  wss 3911  wpss 3912   class class class wbr 5106  ωcom 7803  cen 8883  Fincfn 8886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-om 7804  df-en 8887  df-fin 8890
This theorem is referenced by: (None)
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