Theorem bdssexi 16673

Description: Bounded version of ssexi 4248. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdssexi.bd	⊢ BOUNDED 𝐴
bdssexi.1	⊢ 𝐵 ∈ V
bdssexi.2	⊢ 𝐴 ⊆ 𝐵

Assertion

Ref	Expression
bdssexi	⊢ 𝐴 ∈ V

Proof of Theorem bdssexi

Step	Hyp	Ref	Expression
1		bdssexi.2	. 2 ⊢ 𝐴 ⊆ 𝐵
2		bdssexi.bd	. . 3 ⊢ BOUNDED 𝐴
3		bdssexi.1	. . 3 ⊢ 𝐵 ∈ V
4	2, 3	bdssex 16672	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
5	1, 4	ax-mp 5	1 ⊢ 𝐴 ∈ V

Syntax hints:   ∈ wcel 2203  Vcvv 2813   ⊆ wss 3211  BOUNDED wbdc 16610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-bdsep 16654

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224  df-bdc 16611

This theorem is referenced by: (None)