Theorem bj-adjg1 36228

Description: Existence of the result of the adjunction (generalized only in the first term since this suffices for current applications). (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-adjg1	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ {𝑥}) ∈ V)

Proof of Theorem bj-adjg1

Dummy variables 𝑦 𝑧 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uneq1 4156	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∪ {𝑥}) = (𝐴 ∪ {𝑥}))
2	1	eleq1d 2817	. 2 ⊢ (𝑦 = 𝐴 → ((𝑦 ∪ {𝑥}) ∈ V ↔ (𝐴 ∪ {𝑥}) ∈ V))
3		ax-bj-adj 36227	. . . . 5 ⊢ ∀𝑦∀𝑥∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥))
4	3	spi 2176	. . . 4 ⊢ ∀𝑥∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥))
5	4	spi 2176	. . 3 ⊢ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥))
6		bj-axadj 36226	. . 3 ⊢ ((𝑦 ∪ {𝑥}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥)))
7	5, 6	mpbir 230	. 2 ⊢ (𝑦 ∪ {𝑥}) ∈ V
8	2, 7	vtoclg 3542	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ {𝑥}) ∈ V)

Syntax hints:   → wi 4   ↔ wb 205   ∨ wo 844  ∀wal 1538   = wceq 1540  ∃wex 1780   ∈ wcel 2105  Vcvv 3473   ∪ cun 3946  {csn 4628

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702  ax-bj-adj 36227

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3953  df-sn 4629

This theorem is referenced by:  bj-snfromadj  36229  bj-prfromadj  36230