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Theorem snfiOLD 9084
Description: Obsolete version of snfi 9083 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 8678 . . . 4 1o ∈ ω
2 ensn1g 9062 . . . 4 (𝐴 ∈ V → {𝐴} ≈ 1o)
3 breq2 5147 . . . . 5 (𝑥 = 1o → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ 1o))
43rspcev 3622 . . . 4 ((1o ∈ ω ∧ {𝐴} ≈ 1o) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
51, 2, 4sylancr 587 . . 3 (𝐴 ∈ V → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
6 snprc 4717 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
7 en0 9058 . . . . 5 ({𝐴} ≈ ∅ ↔ {𝐴} = ∅)
8 peano1 7910 . . . . . 6 ∅ ∈ ω
9 breq2 5147 . . . . . . 7 (𝑥 = ∅ → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ ∅))
109rspcev 3622 . . . . . 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 9016 . 2 ({𝐴} ∈ Fin ↔ ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
1614, 15mpbir 231 1 {𝐴} ∈ Fin
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  c0 4333  {csn 4626   class class class wbr 5143  ωcom 7887  1oc1o 8499  cen 8982  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-om 7888  df-1o 8506  df-en 8986  df-fin 8989
This theorem is referenced by: (None)
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