Theorem snfiOLD 9110

Description: Obsolete version of snfi 9109 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.

Step	Hyp	Ref	Expression
1		1onn 8696	. . . 4 ⊢ 1o ∈ ω
2		ensn1g 9084	. . . 4 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o)
3		breq2 5170	. . . . 5 ⊢ (𝑥 = 1o → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ 1o))
4	3	rspcev 3635	. . . 4 ⊢ ((1o ∈ ω ∧ {𝐴} ≈ 1o) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
5	1, 2, 4	sylancr 586	. . 3 ⊢ (𝐴 ∈ V → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
6		snprc 4742	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
7		en0 9078	. . . . 5 ⊢ ({𝐴} ≈ ∅ ↔ {𝐴} = ∅)
8		peano1 7927	. . . . . 6 ⊢ ∅ ∈ ω
9		breq2 5170	. . . . . . 7 ⊢ (𝑥 = ∅ → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ ∅))
10	9	rspcev 3635	. . . . . 6 ⊢ ((∅ ∈ ω ∧ {𝐴} ≈ ∅) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
11	8, 10	mpan 689	. . . . 5 ⊢ ({𝐴} ≈ ∅ → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
12	7, 11	sylbir 235	. . . 4 ⊢ ({𝐴} = ∅ → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
13	6, 12	sylbi 217	. . 3 ⊢ (¬ 𝐴 ∈ V → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
14	5, 13	pm2.61i 182	. 2 ⊢ ∃𝑥 ∈ ω {𝐴} ≈ 𝑥
15		isfi 9036	. 2 ⊢ ({𝐴} ∈ Fin ↔ ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
16	14, 15	mpbir 231	1 ⊢ {𝐴} ∈ Fin

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488  ∅c0 4352  {csn 4648   class class class wbr 5166  ωcom 7903  1_oc1o 8515   ≈ cen 9000  Fincfn 9003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-om 7904  df-1o 8522  df-en 9004  df-fin 9007

This theorem is referenced by: (None)