Theorem bj-axadj 37564

Description: Two ways of stating the axiom of adjunction (which is the universal closure of either side, see ax-bj-adj 37565). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axadj	⊢ ((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦)))

Distinct variable groups:   𝑥,𝑧,𝑡   𝑦,𝑧,𝑡

Proof of Theorem bj-axadj

Step	Hyp	Ref	Expression
1		elun 4115	. . 3 ⊢ (𝑡 ∈ (𝑥 ∪ {𝑦}) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ {𝑦}))
2		velsn 4610	. . . 4 ⊢ (𝑡 ∈ {𝑦} ↔ 𝑡 = 𝑦)
3	2	orbi2i 925	. . 3 ⊢ ((𝑡 ∈ 𝑥 ∨ 𝑡 ∈ {𝑦}) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))
4	1, 3	bitri 278	. 2 ⊢ (𝑡 ∈ (𝑥 ∪ {𝑦}) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))
5	4	bj-clex 37554	1 ⊢ ((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦)))

Syntax hints:   ↔ wb 209   ∨ wo 860  ∀wal 1565  ∃wex 1806   ∈ wcel 2149  Vcvv 3463   ∪ cun 3911  {csn 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-sn 4595

This theorem is referenced by:  bj-adjg1  37566