Theorem bj-unexg 15413

Description: unexg 4474 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-unexg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)

Proof of Theorem bj-unexg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uneq1 3306	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
2		eleq1 2256	. . 3 ⊢ ((𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦) → ((𝑥 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝑦) ∈ V))
3	1, 2	syl 14	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝑦) ∈ V))
4		uneq2 3307	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
5		eleq1 2256	. . 3 ⊢ ((𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵) → ((𝐴 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝐵) ∈ V))
6	4, 5	syl 14	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝐵) ∈ V))
7		vex 2763	. . 3 ⊢ 𝑥 ∈ V
8		vex 2763	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	bj-unex 15411	. 2 ⊢ (𝑥 ∪ 𝑦) ∈ V
10	3, 6, 9	vtocl2g 2824	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ∪ cun 3151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-pr 4238  ax-un 4464  ax-bd0 15305  ax-bdor 15308  ax-bdex 15311  ax-bdeq 15312  ax-bdel 15313  ax-bdsb 15314  ax-bdsep 15376

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-uni 3836  df-bdc 15333

This theorem is referenced by:  bj-sucexg  15414