Theorem bdssexg 10024

Description: Bounded version of ssexg 3896. (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 2967	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝐵))
2	1	imbi1d 220	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ⊆ 𝑥 → 𝐴 ∈ V) ↔ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)))
3		bdssexg.bd	. . . 4 ⊢ BOUNDED 𝐴
4		vex 2560	. . . 4 ⊢ 𝑥 ∈ V
5	3, 4	bdssex 10022	. . 3 ⊢ (𝐴 ⊆ 𝑥 → 𝐴 ∈ V)
6	2, 5	vtoclg 2613	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ⊆ 𝐵 → 𝐴 ∈ V))
7	6	impcom 116	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-bdc 9961

This theorem is referenced by:  bdssexd  10025  bdrabexg  10026  bdunexb  10040