Theorem bdssexd 16667

Description: Bounded version of ssexd 4249. (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 16666	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
5	1, 2, 4	syl2anc 411	1 ⊢ (𝜑 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2203  Vcvv 2812   ⊆ wss 3210  BOUNDED wbdc 16602

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-bdsep 16646

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-in 3216  df-ss 3223  df-bdc 16603

This theorem is referenced by: (None)