Theorem bdssexd 15551

Description: Bounded version of ssexd 4173. (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 15550	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
5	1, 2, 4	syl2anc 411	1 ⊢ (𝜑 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2167  Vcvv 2763   ⊆ wss 3157  BOUNDED wbdc 15486

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-bdsep 15530

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-bdc 15487

This theorem is referenced by: (None)