Theorem bdssex 15007

Description: Bounded version of ssex 4152. (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 3154	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
2		bdssex.bd	. . . 4 ⊢ BOUNDED 𝐴
3		bdssex.1	. . . 4 ⊢ 𝐵 ∈ V
4	2, 3	bdinex2 15005	. . 3 ⊢ (𝐴 ∩ 𝐵) ∈ V
5		eleq1 2250	. . 3 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → ((𝐴 ∩ 𝐵) ∈ V ↔ 𝐴 ∈ V))
6	4, 5	mpbii 148	. 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → 𝐴 ∈ V)
7	1, 6	sylbi 121	1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1363   ∈ wcel 2158  Vcvv 2749   ∩ cin 3140   ⊆ wss 3141  BOUNDED wbdc 14945

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-bdsep 14989

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-in 3147  df-ss 3154  df-bdc 14946

This theorem is referenced by:  bdssexi  15008  bdssexg  15009  bdfind  15051