Theorem bdunexb 15720

Description: Bounded version of unexb 4487. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdunex.bd1	⊢ BOUNDED 𝐴
bdunex.bd2	⊢ BOUNDED 𝐵

Assertion

Ref	Expression
bdunexb	⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)

Proof of Theorem bdunexb

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

Step	Hyp	Ref	Expression
1		uneq1 3319	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝑦))
2	1	eleq1d 2273	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝑦) ∈ V))
3		uneq2 3320	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 ∪ 𝑦) = (𝐴 ∪ 𝐵))
4	3	eleq1d 2273	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∪ 𝑦) ∈ V ↔ (𝐴 ∪ 𝐵) ∈ V))
5		vex 2774	. . . 4 ⊢ 𝑥 ∈ V
6		vex 2774	. . . 4 ⊢ 𝑦 ∈ V
7	5, 6	bj-unex 15719	. . 3 ⊢ (𝑥 ∪ 𝑦) ∈ V
8	2, 4, 7	vtocl2g 2836	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
9		ssun1 3335	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
10		bdunex.bd1	. . . . 5 ⊢ BOUNDED 𝐴
11	10	bdssexg 15704	. . . 4 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
12	9, 11	mpan 424	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐴 ∈ V)
13		ssun2 3336	. . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
14		bdunex.bd2	. . . . 5 ⊢ BOUNDED 𝐵
15	14	bdssexg 15704	. . . 4 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V)
16	13, 15	mpan 424	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ V → 𝐵 ∈ V)
17	12, 16	jca 306	. 2 ⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
18	8, 17	impbii 126	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1372   ∈ wcel 2175  Vcvv 2771   ∪ cun 3163   ⊆ wss 3165  BOUNDED wbdc 15640

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-pr 4252  ax-un 4478  ax-bd0 15613  ax-bdor 15616  ax-bdex 15619  ax-bdeq 15620  ax-bdel 15621  ax-bdsb 15622  ax-bdsep 15684

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-sn 3638  df-pr 3639  df-uni 3850  df-bdc 15641

This theorem is referenced by: (None)