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Theorem bj-axadj 37042
Description: Two ways of stating the axiom of adjunction (which is the universal closure of either side, see ax-bj-adj 37043). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axadj ((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡 = 𝑦)))
Distinct variable groups:   𝑥,𝑧,𝑡   𝑦,𝑧,𝑡

Proof of Theorem bj-axadj
StepHypRef Expression
1 elun 4153 . . 3 (𝑡 ∈ (𝑥 ∪ {𝑦}) ↔ (𝑡𝑥𝑡 ∈ {𝑦}))
2 velsn 4642 . . . 4 (𝑡 ∈ {𝑦} ↔ 𝑡 = 𝑦)
32orbi2i 913 . . 3 ((𝑡𝑥𝑡 ∈ {𝑦}) ↔ (𝑡𝑥𝑡 = 𝑦))
41, 3bitri 275 . 2 (𝑡 ∈ (𝑥 ∪ {𝑦}) ↔ (𝑡𝑥𝑡 = 𝑦))
54bj-clex 37032 1 ((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wo 848  wal 1538  wex 1779  wcel 2108  Vcvv 3480  cun 3949  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-sn 4627
This theorem is referenced by:  bj-adjg1  37044
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