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Theorem iunn0m 3717
Description: There is an inhabited class in an indexed collection 𝐵(𝑥) iff the indexed union of them is inhabited. (Contributed by Jim Kingdon, 16-Aug-2018.)
Assertion
Ref Expression
iunn0m (∃𝑥𝐴𝑦 𝑦𝐵 ↔ ∃𝑦 𝑦 𝑥𝐴 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunn0m
StepHypRef Expression
1 rexcom4 2577 . 2 (∃𝑥𝐴𝑦 𝑦𝐵 ↔ ∃𝑦𝑥𝐴 𝑦𝐵)
2 eliun 3661 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
32exbii 1496 . 2 (∃𝑦 𝑦 𝑥𝐴 𝐵 ↔ ∃𝑦𝑥𝐴 𝑦𝐵)
41, 3bitr4i 176 1 (∃𝑥𝐴𝑦 𝑦𝐵 ↔ ∃𝑦 𝑦 𝑥𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wb 98  wex 1381  wcel 1393  wrex 2307   ciun 3657
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-iun 3659
This theorem is referenced by: (None)
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