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Theorem snfiOLD 9082
Description: Obsolete version of snfi 9081 as of 13-Jan-2025. (Contributed by NM, 4-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snfiOLD {𝐴} ∈ Fin

Proof of Theorem snfiOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 1onn 8676 . . . 4 1o ∈ ω
2 ensn1g 9060 . . . 4 (𝐴 ∈ V → {𝐴} ≈ 1o)
3 breq2 5151 . . . . 5 (𝑥 = 1o → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ 1o))
43rspcev 3621 . . . 4 ((1o ∈ ω ∧ {𝐴} ≈ 1o) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
51, 2, 4sylancr 587 . . 3 (𝐴 ∈ V → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
6 snprc 4721 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
7 en0 9056 . . . . 5 ({𝐴} ≈ ∅ ↔ {𝐴} = ∅)
8 peano1 7910 . . . . . 6 ∅ ∈ ω
9 breq2 5151 . . . . . . 7 (𝑥 = ∅ → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ ∅))
109rspcev 3621 . . . . . 6 ((∅ ∈ ω ∧ {𝐴} ≈ ∅) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
118, 10mpan 690 . . . . 5 ({𝐴} ≈ ∅ → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
127, 11sylbir 235 . . . 4 ({𝐴} = ∅ → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
136, 12sylbi 217 . . 3 𝐴 ∈ V → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
145, 13pm2.61i 182 . 2 𝑥 ∈ ω {𝐴} ≈ 𝑥
15 isfi 9014 . 2 ({𝐴} ∈ Fin ↔ ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
1614, 15mpbir 231 1 {𝐴} ∈ Fin
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1536  wcel 2105  wrex 3067  Vcvv 3477  c0 4338  {csn 4630   class class class wbr 5147  ωcom 7886  1oc1o 8497  cen 8980  Fincfn 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-mo 2537  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-om 7887  df-1o 8504  df-en 8984  df-fin 8987
This theorem is referenced by: (None)
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