Theorem snfiOLD 8976

Description: Obsolete version of snfi 8975 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 8565	. . . 4 ⊢ 1o ∈ ω
2		ensn1g 8954	. . . 4 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o)
3		breq2 5099	. . . . 5 ⊢ (𝑥 = 1o → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ 1o))
4	3	rspcev 3579	. . . 4 ⊢ ((1o ∈ ω ∧ {𝐴} ≈ 1o) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
5	1, 2, 4	sylancr 587	. . 3 ⊢ (𝐴 ∈ V → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
6		snprc 4671	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
7		en0 8950	. . . . 5 ⊢ ({𝐴} ≈ ∅ ↔ {𝐴} = ∅)
8		peano1 7829	. . . . . 6 ⊢ ∅ ∈ ω
9		breq2 5099	. . . . . . 7 ⊢ (𝑥 = ∅ → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ ∅))
10	9	rspcev 3579	. . . . . 6 ⊢ ((∅ ∈ ω ∧ {𝐴} ≈ ∅) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
11	8, 10	mpan 690	. . . . 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 8908	. 2 ⊢ ({𝐴} ∈ Fin ↔ ∃𝑥 ∈ ω {𝐴} ≈ 𝑥)
16	14, 15	mpbir 231	1 ⊢ {𝐴} ∈ Fin

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3438  ∅c0 4286  {csn 4579   class class class wbr 5095  ωcom 7806  1_oc1o 8388   ≈ cen 8876  Fincfn 8879

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 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-om 7807  df-1o 8395  df-en 8880  df-fin 8883

This theorem is referenced by: (None)