Theorem bdssex 16845

Description: Bounded version of ssex 4268. (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 3233	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
2		bdssex.bd	. . . 4 ⊢ BOUNDED 𝐴
3		bdssex.1	. . . 4 ⊢ 𝐵 ∈ V
4	2, 3	bdinex2 16843	. . 3 ⊢ (𝐴 ∩ 𝐵) ∈ V
5		eleq1 2301	. . 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 1402   ∈ wcel 2209  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220  BOUNDED wbdc 16783

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-bdsep 16827

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-bdc 16784

This theorem is referenced by:  bdssexi  16846  bdssexg  16847  bdfind  16889