Theorem bj-adjg1 37403

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 4098	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∪ {𝑥}) = (𝐴 ∪ {𝑥}))
2	1	eleq1d 2825	. 2 ⊢ (𝑦 = 𝐴 → ((𝑦 ∪ {𝑥}) ∈ V ↔ (𝐴 ∪ {𝑥}) ∈ V))
3		ax-bj-adj 37402	. . . . 5 ⊢ ∀𝑦∀𝑥∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥))
4	3	spi 2196	. . . 4 ⊢ ∀𝑥∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥))
5	4	spi 2196	. . 3 ⊢ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥))
6		bj-axadj 37401	. . 3 ⊢ ((𝑦 ∪ {𝑥}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥)))
7	5, 6	mpbir 232	. 2 ⊢ (𝑦 ∪ {𝑥}) ∈ V
8	2, 7	vtoclg 3502	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ {𝑥}) ∈ V)

Syntax hints:   → wi 4   ↔ wb 207   ∨ wo 853  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3432   ∪ cun 3888  {csn 4562

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-12 2189  ax-ext 2712  ax-bj-adj 37402

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 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-un 3895  df-sn 4563

This theorem is referenced by:  bj-snfromadj  37404  bj-prfromadj  37405