Theorem bj-axadj 37185

Description: Two ways of stating the axiom of adjunction (which is the universal closure of either side, see ax-bj-adj 37186). (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 4103	. . 3 ⊢ (𝑡 ∈ (𝑥 ∪ {𝑦}) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ {𝑦}))
2		velsn 4594	. . . 4 ⊢ (𝑡 ∈ {𝑦} ↔ 𝑡 = 𝑦)
3	2	orbi2i 912	. . 3 ⊢ ((𝑡 ∈ 𝑥 ∨ 𝑡 ∈ {𝑦}) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))
4	1, 3	bitri 275	. 2 ⊢ (𝑡 ∈ (𝑥 ∪ {𝑦}) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))
5	4	bj-clex 37175	1 ⊢ ((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦)))

Syntax hints:   ↔ wb 206   ∨ wo 847  ∀wal 1539  ∃wex 1780   ∈ wcel 2113  Vcvv 3438   ∪ cun 3897  {csn 4578

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-un 3904  df-sn 4579

This theorem is referenced by:  bj-adjg1  37187