Theorem bdssex 10693

Description: Bounded version of ssex 3915. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdssex.bd	⊢ BOUNDED 𝐴
bdssex.1	⊢ 𝐵 ∈ V

Assertion

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

Proof of Theorem bdssex

Step	Hyp	Ref	Expression
1		df-ss 2986	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
2		bdssex.bd	. . . 4 ⊢ BOUNDED 𝐴
3		bdssex.1	. . . 4 ⊢ 𝐵 ∈ V
4	2, 3	bdinex2 10691	. . 3 ⊢ (𝐴 ∩ 𝐵) ∈ V
5		eleq1 2141	. . 3 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → ((𝐴 ∩ 𝐵) ∈ V ↔ 𝐴 ∈ V))
6	4, 5	mpbii 146	. 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → 𝐴 ∈ V)
7	1, 6	sylbi 119	1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ∩ cin 2972   ⊆ wss 2973  BOUNDED wbdc 10631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bdsep 10675

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-bdc 10632

This theorem is referenced by:  bdssexi  10694  bdssexg  10695  bdfind  10741