Theorem bdssex 15394

Description: Bounded version of ssex 4166. (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 3166	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
2		bdssex.bd	. . . 4 ⊢ BOUNDED 𝐴
3		bdssex.1	. . . 4 ⊢ 𝐵 ∈ V
4	2, 3	bdinex2 15392	. . 3 ⊢ (𝐴 ∩ 𝐵) ∈ V
5		eleq1 2256	. . 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 1364   ∈ wcel 2164  Vcvv 2760   ∩ cin 3152   ⊆ wss 3153  BOUNDED wbdc 15332

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 2175  ax-bdsep 15376

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-bdc 15333

This theorem is referenced by:  bdssexi  15395  bdssexg  15396  bdfind  15438