Theorem ssnnfiOLD 9193

Description: Obsolete version of ssnnfi 9192 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 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)

Proof of Theorem ssnnfiOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sspss 4091	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ⊊ 𝐴 ∨ 𝐵 = 𝐴))
2		pssnn 9191	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)
3		elnn 7879	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑥 ∈ ω)
4	3	expcom 412	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑥 ∈ 𝐴 → 𝑥 ∈ ω))
5	4	anim1d 609	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((𝑥 ∈ 𝐴 ∧ 𝐵 ≈ 𝑥) → (𝑥 ∈ ω ∧ 𝐵 ≈ 𝑥)))
6	5	reximdv2 3154	. . . . . 6 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥 → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥))
7	6	adantr 479	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → (∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥 → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥))
8	2, 7	mpd 15	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)
9		eleq1 2813	. . . . . 6 ⊢ (𝐵 = 𝐴 → (𝐵 ∈ ω ↔ 𝐴 ∈ ω))
10	9	biimparc 478	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐵 ∈ ω)
11		enrefnn 9070	. . . . 5 ⊢ (𝐵 ∈ ω → 𝐵 ≈ 𝐵)
12		breq2 5147	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐵 ≈ 𝑥 ↔ 𝐵 ≈ 𝐵))
13	12	rspcev 3601	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐵 ≈ 𝐵) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)
14	10, 11, 13	syl2anc2 583	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)
15	8, 14	jaodan 955	. . 3 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ⊊ 𝐴 ∨ 𝐵 = 𝐴)) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)
16	1, 15	sylan2b 592	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)
17		isfi 8995	. 2 ⊢ (𝐵 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)
18	16, 17	sylibr 233	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098  ∃wrex 3060   ⊆ wss 3939   ⊊ wpss 3940   class class class wbr 5143  ωcom 7868   ≈ cen 8959  Fincfn 8962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-om 7869  df-en 8963  df-fin 8966

This theorem is referenced by: (None)