Theorem bj-axadj 37394

Description: Two ways of stating the axiom of adjunction (which is the universal closure of either side, see ax-bj-adj 37395). (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 4083	. . 3 ⊢ (𝑡 ∈ (𝑥 ∪ {𝑦}) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ {𝑦}))
2		velsn 4571	. . . 4 ⊢ (𝑡 ∈ {𝑦} ↔ 𝑡 = 𝑦)
3	2	orbi2i 918	. . 3 ⊢ ((𝑡 ∈ 𝑥 ∨ 𝑡 ∈ {𝑦}) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))
4	1, 3	bitri 276	. 2 ⊢ (𝑡 ∈ (𝑥 ∪ {𝑦}) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))
5	4	bj-clex 37384	1 ⊢ ((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦)))

Syntax hints:   ↔ wb 207   ∨ wo 853  ∀wal 1545  ∃wex 1786   ∈ wcel 2119  Vcvv 3431   ∪ cun 3881  {csn 4555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-un 3888  df-sn 4556

This theorem is referenced by:  bj-adjg1  37396