Theorem bj-uniexg 13116

Description: uniexg 4361 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 3745	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	eleq1d 2208	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
3		vex 2689	. . 3 ⊢ 𝑥 ∈ V
4	3	bj-uniex 13115	. 2 ⊢ ∪ 𝑥 ∈ V
5	2, 4	vtoclg 2746	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  Vcvv 2686  ∪ cuni 3736

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-un 4355  ax-bd0 13011  ax-bdex 13017  ax-bdel 13019  ax-bdsb 13020  ax-bdsep 13082

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-uni 3737  df-bdc 13039

This theorem is referenced by: (None)