Theorem bdssexg 16930

Description: Bounded version of ssexg 4272. (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 3272	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝐵))
2	1	imbi1d 231	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ⊆ 𝑥 → 𝐴 ∈ V) ↔ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)))
3		bdssexg.bd	. . . 4 ⊢ BOUNDED 𝐴
4		vex 2824	. . . 4 ⊢ 𝑥 ∈ V
5	3, 4	bdssex 16928	. . 3 ⊢ (𝐴 ⊆ 𝑥 → 𝐴 ∈ V)
6	2, 5	vtoclg 2883	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ∈ V))
7	6	impcom 125	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  BOUNDED wbdc 16866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-bdsep 16910

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-bdc 16867

This theorem is used by:  bdssexd  16931  bdrabexg  16932  bdunexb  16946