Theorem ssexOLD 5293

Description: Obsolete version of ssex 5292 as of 18-Jul-2026. (Contributed by NM, 27-Apr-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ssex.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
ssexOLD	⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)

Proof of Theorem ssexOLD

Step	Hyp	Ref	Expression
1		dfss2 3924	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
2		ssex.1	. . . 4 ⊢ 𝐵 ∈ V
3	2	inex2 5288	. . 3 ⊢ (𝐴 ∩ 𝐵) ∈ V
4		eleq1 2851	. . 3 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → ((𝐴 ∩ 𝐵) ∈ V ↔ 𝐴 ∈ V))
5	3, 4	mpbii 236	. 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → 𝐴 ∈ V)
6	1, 5	sylbi 220	1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∩ cin 3905   ⊆ wss 3906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-in 3913  df-ss 3923

This theorem is referenced by: (None)