Theorem bdssexi 15116

Description: Bounded version of ssexi 4156. (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 15115	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
5	1, 4	ax-mp 5	1 ⊢ 𝐴 ∈ V

Syntax hints:   ∈ wcel 2160  Vcvv 2752   ⊆ wss 3144  BOUNDED wbdc 15053

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bdsep 15097

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-bdc 15054

This theorem is referenced by: (None)