Theorem bdssexi 16498

Description: Bounded version of ssexi 4227. (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 16497	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
5	1, 4	ax-mp 5	1 ⊢ 𝐴 ∈ V

Syntax hints:   ∈ wcel 2202  Vcvv 2802   ⊆ wss 3200  BOUNDED wbdc 16435

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-bdsep 16479

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-bdc 16436

This theorem is referenced by: (None)