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Theorem bdunexb 10978
Description: Bounded version of unexb 4223. (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) ↔ (𝐴𝐵) ∈ V)

Proof of Theorem bdunexb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 3129 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
21eleq1d 2151 . . 3 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
3 uneq2 3130 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
43eleq1d 2151 . . 3 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
5 vex 2613 . . . 4 𝑥 ∈ V
6 vex 2613 . . . 4 𝑦 ∈ V
75, 6bj-unex 10977 . . 3 (𝑥𝑦) ∈ V
82, 4, 7vtocl2g 2671 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
9 ssun1 3145 . . . 4 𝐴 ⊆ (𝐴𝐵)
10 bdunex.bd1 . . . . 5 BOUNDED 𝐴
1110bdssexg 10962 . . . 4 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐴 ∈ V)
129, 11mpan 415 . . 3 ((𝐴𝐵) ∈ V → 𝐴 ∈ V)
13 ssun2 3146 . . . 4 𝐵 ⊆ (𝐴𝐵)
14 bdunex.bd2 . . . . 5 BOUNDED 𝐵
1514bdssexg 10962 . . . 4 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐵 ∈ V)
1613, 15mpan 415 . . 3 ((𝐴𝐵) ∈ V → 𝐵 ∈ V)
1712, 16jca 300 . 2 ((𝐴𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
188, 17impbii 124 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1285  wcel 1434  Vcvv 2610  cun 2980  wss 2982  BOUNDED wbdc 10898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-pr 3992  ax-un 4216  ax-bd0 10871  ax-bdor 10874  ax-bdex 10877  ax-bdeq 10878  ax-bdel 10879  ax-bdsb 10880  ax-bdsep 10942
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-uni 3622  df-bdc 10899
This theorem is referenced by: (None)
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