Theorem bdssexd 13747

Description: Bounded version of ssexd 4121. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdssexd.1	⊢ (𝜑 → 𝐵 ∈ 𝐶)
bdssexd.2	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
bdssexd.bd	⊢ BOUNDED 𝐴

Assertion

Ref	Expression
bdssexd	⊢ (𝜑 → 𝐴 ∈ V)

Proof of Theorem bdssexd

Step	Hyp	Ref	Expression
1		bdssexd.2	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		bdssexd.1	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
3		bdssexd.bd	. . 3 ⊢ BOUNDED 𝐴
4	3	bdssexg 13746	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
5	1, 2, 4	syl2anc 409	1 ⊢ (𝜑 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2136  Vcvv 2725   ⊆ wss 3115  BOUNDED wbdc 13682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-bdsep 13726

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-in 3121  df-ss 3128  df-bdc 13683

This theorem is referenced by: (None)