Theorem bj-adjg1 37660

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 4116	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∪ {𝑥}) = (𝐴 ∪ {𝑥}))
2	1	eleq1d 2848	. 2 ⊢ (𝑦 = 𝐴 → ((𝑦 ∪ {𝑥}) ∈ V ↔ (𝐴 ∪ {𝑥}) ∈ V))
3		ax-bj-adj 37659	. . . . 5 ⊢ ∀𝑦∀𝑥∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥))
4	3	spi 2220	. . . 4 ⊢ ∀𝑥∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥))
5	4	spi 2220	. . 3 ⊢ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥))
6		bj-axadj 37658	. . 3 ⊢ ((𝑦 ∪ {𝑥}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥)))
7	5, 6	mpbir 234	. 2 ⊢ (𝑦 ∪ {𝑥}) ∈ V
8	2, 7	vtoclg 3523	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ {𝑥}) ∈ V)

Syntax hints:   → wi 4   ↔ wb 209   ∨ wo 860  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455   ∪ cun 3904  {csn 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735  ax-bj-adj 37659

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-un 3911  df-sn 4591

This theorem is referenced by:  bj-snfromadj  37661  bj-prfromadj  37662