Theorem bdssexg 13786

Description: Bounded version of ssexg 4121. (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 3166	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝐵))
2	1	imbi1d 230	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ⊆ 𝑥 → 𝐴 ∈ V) ↔ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)))
3		bdssexg.bd	. . . 4 ⊢ BOUNDED 𝐴
4		vex 2729	. . . 4 ⊢ 𝑥 ∈ V
5	3, 4	bdssex 13784	. . 3 ⊢ (𝐴 ⊆ 𝑥 → 𝐴 ∈ V)
6	2, 5	vtoclg 2786	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ∈ V))
7	6	impcom 124	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  BOUNDED wbdc 13722

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 13766

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 2297  df-v 2728  df-in 3122  df-ss 3129  df-bdc 13723

This theorem is referenced by:  bdssexd  13787  bdrabexg  13788  bdunexb  13802