Theorem bj-adjg1 37009

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 4184	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∪ {𝑥}) = (𝐴 ∪ {𝑥}))
2	1	eleq1d 2829	. 2 ⊢ (𝑦 = 𝐴 → ((𝑦 ∪ {𝑥}) ∈ V ↔ (𝐴 ∪ {𝑥}) ∈ V))
3		ax-bj-adj 37008	. . . . 5 ⊢ ∀𝑦∀𝑥∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥))
4	3	spi 2185	. . . 4 ⊢ ∀𝑥∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥))
5	4	spi 2185	. . 3 ⊢ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥))
6		bj-axadj 37007	. . 3 ⊢ ((𝑦 ∪ {𝑥}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥)))
7	5, 6	mpbir 231	. 2 ⊢ (𝑦 ∪ {𝑥}) ∈ V
8	2, 7	vtoclg 3566	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ {𝑥}) ∈ V)

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 846  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-bj-adj 37008

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-sn 4649

This theorem is referenced by:  bj-snfromadj  37010  bj-prfromadj  37011