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Theorem bj-adjg1 37026
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 (𝐴𝑉 → (𝐴 ∪ {𝑥}) ∈ V)

Proof of Theorem bj-adjg1
Dummy variables 𝑦 𝑧 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 4171 . . 3 (𝑦 = 𝐴 → (𝑦 ∪ {𝑥}) = (𝐴 ∪ {𝑥}))
21eleq1d 2824 . 2 (𝑦 = 𝐴 → ((𝑦 ∪ {𝑥}) ∈ V ↔ (𝐴 ∪ {𝑥}) ∈ V))
3 ax-bj-adj 37025 . . . . 5 𝑦𝑥𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑦𝑡 = 𝑥))
43spi 2182 . . . 4 𝑥𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑦𝑡 = 𝑥))
54spi 2182 . . 3 𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑦𝑡 = 𝑥))
6 bj-axadj 37024 . . 3 ((𝑦 ∪ {𝑥}) ∈ V ↔ ∃𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑦𝑡 = 𝑥)))
75, 6mpbir 231 . 2 (𝑦 ∪ {𝑥}) ∈ V
82, 7vtoclg 3554 1 (𝐴𝑉 → (𝐴 ∪ {𝑥}) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wo 847  wal 1535   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cun 3961  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-bj-adj 37025
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632
This theorem is referenced by:  bj-snfromadj  37027  bj-prfromadj  37028
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