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Theorem unielid 43121
Description: Two ways to say the union of a class is an element of that class. (Contributed by RP, 27-Jan-2025.)
Assertion
Ref Expression
unielid ( 𝐴𝐴 ↔ ∃𝑥𝐴𝑦𝐴 𝑦𝑥)
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem unielid
StepHypRef Expression
1 ssid 4025 . 2 𝐴𝐴
2 unielss 43120 . 2 (𝐴𝐴 → ( 𝐴𝐴 ↔ ∃𝑥𝐴𝑦𝐴 𝑦𝑥))
31, 2ax-mp 5 1 ( 𝐴𝐴 ↔ ∃𝑥𝐴𝑦𝐴 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2103  wral 3063  wrex 3072  wss 3970   cuni 4931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-ss 3987  df-uni 4932
This theorem is referenced by:  onsupnmax  43130  onsupeqmax  43148  onsupeqnmax  43149
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