Theorem bdssexd 13940

Description: Bounded version of ssexd 4129. (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 13939	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
5	1, 2, 4	syl2anc 409	1 ⊢ (𝜑 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2141  Vcvv 2730   ⊆ wss 3121  BOUNDED wbdc 13875

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-bdsep 13919

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-bdc 13876

This theorem is referenced by: (None)