Theorem ssnnfiOLD 8953

Description: Obsolete version of ssnnfi 8952 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 4034	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ⊊ 𝐴 ∨ 𝐵 = 𝐴))
2		pssnn 8951	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)
3		elnn 7723	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑥 ∈ ω)
4	3	expcom 414	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝑥 ∈ 𝐴 → 𝑥 ∈ ω))
5	4	anim1d 611	. . . . . . 7 ⊢ (𝐴 ∈ ω → ((𝑥 ∈ 𝐴 ∧ 𝐵 ≈ 𝑥) → (𝑥 ∈ ω ∧ 𝐵 ≈ 𝑥)))
6	5	reximdv2 3199	. . . . . 6 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥 → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥))
7	6	adantr 481	. . . . 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 2826	. . . . . 6 ⊢ (𝐵 = 𝐴 → (𝐵 ∈ ω ↔ 𝐴 ∈ ω))
10	9	biimparc 480	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐵 ∈ ω)
11		enrefnn 8837	. . . . 5 ⊢ (𝐵 ∈ ω → 𝐵 ≈ 𝐵)
12		breq2 5078	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐵 ≈ 𝑥 ↔ 𝐵 ≈ 𝐵))
13	12	rspcev 3561	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐵 ≈ 𝐵) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)
14	10, 11, 13	syl2anc2 585	. . . 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 594	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)
17		isfi 8764	. 2 ⊢ (𝐵 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)
18	16, 17	sylibr 233	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ⊆ wss 3887   ⊊ wpss 3888   class class class wbr 5074  ωcom 7712   ≈ cen 8730  Fincfn 8733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-om 7713  df-en 8734  df-fin 8737

This theorem is referenced by: (None)