Theorem bdssexi 10023

Description: Bounded version of ssexi 3895. (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 10022	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
5	1, 4	ax-mp 7	1 ⊢ 𝐴 ∈ V

Syntax hints:   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-bdc 9961

This theorem is referenced by: (None)