Theorem ssnnfiOLD 9236

Description: Obsolete version of ssnnfi 9235 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 4125	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ⊊ 𝐴 ∨ 𝐵 = 𝐴))
2		pssnn 9234	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)
3		elnn 7914	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑥 ∈ ω)
4	3	expcom 413	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑥 ∈ 𝐴 → 𝑥 ∈ ω))
5	4	anim1d 610	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((𝑥 ∈ 𝐴 ∧ 𝐵 ≈ 𝑥) → (𝑥 ∈ ω ∧ 𝐵 ≈ 𝑥)))
6	5	reximdv2 3170	. . . . . 6 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥 → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥))
7	6	adantr 480	. . . . 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 2832	. . . . . 6 ⊢ (𝐵 = 𝐴 → (𝐵 ∈ ω ↔ 𝐴 ∈ ω))
10	9	biimparc 479	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐵 ∈ ω)
11		enrefnn 9113	. . . . 5 ⊢ (𝐵 ∈ ω → 𝐵 ≈ 𝐵)
12		breq2 5170	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐵 ≈ 𝑥 ↔ 𝐵 ≈ 𝐵))
13	12	rspcev 3635	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐵 ≈ 𝐵) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)
14	10, 11, 13	syl2anc2 584	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)
15	8, 14	jaodan 958	. . 3 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ⊊ 𝐴 ∨ 𝐵 = 𝐴)) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)
16	1, 15	sylan2b 593	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)
17		isfi 9036	. 2 ⊢ (𝐵 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)
18	16, 17	sylibr 234	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  ∃wrex 3076   ⊆ wss 3976   ⊊ wpss 3977   class class class wbr 5166  ωcom 7903   ≈ 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-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

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-nf 1782  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-en 9004  df-fin 9007

This theorem is referenced by: (None)