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Theorem uniel 42429
Description: Two ways to say a union is an element of a class. (Contributed by RP, 27-Jan-2025.)
Assertion
Ref Expression
uniel ( 𝐴𝐵 ↔ ∃𝑥𝐵𝑧(𝑧𝑥 ↔ ∃𝑦𝐴 𝑧𝑦))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵
Allowed substitution hints:   𝐵(𝑦,𝑧)

Proof of Theorem uniel
StepHypRef Expression
1 clabel 2880 . 2 ({𝑧 ∣ ∃𝑦𝐴 𝑧𝑦} ∈ 𝐵 ↔ ∃𝑥(𝑥𝐵 ∧ ∀𝑧(𝑧𝑥 ↔ ∃𝑦𝐴 𝑧𝑦)))
2 dfuni2 4910 . . 3 𝐴 = {𝑧 ∣ ∃𝑦𝐴 𝑧𝑦}
32eleq1i 2823 . 2 ( 𝐴𝐵 ↔ {𝑧 ∣ ∃𝑦𝐴 𝑧𝑦} ∈ 𝐵)
4 df-rex 3070 . 2 (∃𝑥𝐵𝑧(𝑧𝑥 ↔ ∃𝑦𝐴 𝑧𝑦) ↔ ∃𝑥(𝑥𝐵 ∧ ∀𝑧(𝑧𝑥 ↔ ∃𝑦𝐴 𝑧𝑦)))
51, 3, 43bitr4i 303 1 ( 𝐴𝐵 ↔ ∃𝑥𝐵𝑧(𝑧𝑥 ↔ ∃𝑦𝐴 𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wal 1538  wex 1780  wcel 2105  {cab 2708  wrex 3069   cuni 4908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rex 3070  df-uni 4909
This theorem is referenced by:  unielss  42430  onsupmaxb  42451
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