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Theorem bj-axadj 37024
Description: Two ways of stating the axiom of adjunction (which is the universal closure of either side, see ax-bj-adj 37025). (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 4163 . . 3 (𝑡 ∈ (𝑥 ∪ {𝑦}) ↔ (𝑡𝑥𝑡 ∈ {𝑦}))
2 velsn 4647 . . . 4 (𝑡 ∈ {𝑦} ↔ 𝑡 = 𝑦)
32orbi2i 912 . . 3 ((𝑡𝑥𝑡 ∈ {𝑦}) ↔ (𝑡𝑥𝑡 = 𝑦))
41, 3bitri 275 . 2 (𝑡 ∈ (𝑥 ∪ {𝑦}) ↔ (𝑡𝑥𝑡 = 𝑦))
54bj-clex 37014 1 ((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wo 847  wal 1535  wex 1776  wcel 2106  Vcvv 3478  cun 3961  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632
This theorem is referenced by:  bj-adjg1  37026
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