Theorem bj-uniexg 9373

Description: uniexg 4141 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-uniexg	⊢ (A ∈ 𝑉 → ∪ A ∈ V)

Proof of Theorem bj-uniexg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 3580	. . 3 ⊢ (x = A → ∪ x = ∪ A)
2	1	eleq1d 2103	. 2 ⊢ (x = A → (∪ x ∈ V ↔ ∪ A ∈ V))
3		vex 2554	. . 3 ⊢ x ∈ V
4	3	bj-uniex 9372	. 2 ⊢ ∪ x ∈ V
5	2, 4	vtoclg 2607	1 ⊢ (A ∈ 𝑉 → ∪ A ∈ V)

Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  Vcvv 2551  ∪ cuni 3571

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-un 4136  ax-bd0 9268  ax-bdex 9274  ax-bdel 9276  ax-bdsb 9277  ax-bdsep 9339

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-uni 3572  df-bdc 9296

This theorem is referenced by: (None)