Theorem bdssexg 10695

Description: Bounded version of ssexg 3917. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bdssexg.bd	⊢ BOUNDED 𝐴

Assertion

Ref	Expression
bdssexg	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

Proof of Theorem bdssexg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3021	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝐵))
2	1	imbi1d 229	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ⊆ 𝑥 → 𝐴 ∈ V) ↔ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)))
3		bdssexg.bd	. . . 4 ⊢ BOUNDED 𝐴
4		vex 2604	. . . 4 ⊢ 𝑥 ∈ V
5	3, 4	bdssex 10693	. . 3 ⊢ (𝐴 ⊆ 𝑥 → 𝐴 ∈ V)
6	2, 5	vtoclg 2658	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ∈ V))
7	6	impcom 123	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ⊆ wss 2973  BOUNDED wbdc 10631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bdsep 10675

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-bdc 10632

This theorem is referenced by:  bdssexd  10696  bdrabexg  10697  bdunexb  10711