Theorem axprg 5408

Description: Derive The Axiom of Pairing with class variables. (Contributed by GG, 6-Mar-2026.)

Assertion

Ref	Expression
axprg	⊢ ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)

Distinct variable groups:   𝑧,𝐴,𝑤   𝑧,𝐵,𝑤

Proof of Theorem axprg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2765	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝐴 ↔ 𝑥 = 𝐴))
2		eqeq1 2765	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝐵 ↔ 𝑥 = 𝐵))
3	1, 2	orbi12d 931	. . . 4 ⊢ (𝑤 = 𝑥 → ((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)))
4	3	cbvexvw 2065	. . 3 ⊢ (∃𝑤(𝑤 = 𝐴 ∨ 𝑤 = 𝐵) ↔ ∃𝑥(𝑥 = 𝐴 ∨ 𝑥 = 𝐵))
5		axprglem 5407	. . . . 5 ⊢ (𝑥 = 𝐴 → ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
6		axprglem 5407	. . . . . 6 ⊢ (𝑥 = 𝐵 → ∃𝑧∀𝑤((𝑤 = 𝐵 ∨ 𝑤 = 𝐴) → 𝑤 ∈ 𝑧))
7		pm1.4 882	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → (𝑤 = 𝐵 ∨ 𝑤 = 𝐴))
8	7	imim1i 64	. . . . . . . 8 ⊢ (((𝑤 = 𝐵 ∨ 𝑤 = 𝐴) → 𝑤 ∈ 𝑧) → ((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
9	8	alimi 1839	. . . . . . 7 ⊢ (∀𝑤((𝑤 = 𝐵 ∨ 𝑤 = 𝐴) → 𝑤 ∈ 𝑧) → ∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
10	9	eximi 1863	. . . . . 6 ⊢ (∃𝑧∀𝑤((𝑤 = 𝐵 ∨ 𝑤 = 𝐴) → 𝑤 ∈ 𝑧) → ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
11	6, 10	syl 18	. . . . 5 ⊢ (𝑥 = 𝐵 → ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
12	5, 11	jaoi 870	. . . 4 ⊢ ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
13	12	exlimiv 1958	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
14	4, 13	sylbi 220	. 2 ⊢ (∃𝑤(𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
15		alnex 1809	. . . . 5 ⊢ (∀𝑤 ¬ (𝑤 = 𝐴 ∨ 𝑤 = 𝐵) ↔ ¬ ∃𝑤(𝑤 = 𝐴 ∨ 𝑤 = 𝐵))
16		pm2.21 124	. . . . . 6 ⊢ (¬ (𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → ((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
17	16	alimi 1839	. . . . 5 ⊢ (∀𝑤 ¬ (𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → ∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
18	15, 17	sylbir 238	. . . 4 ⊢ (¬ ∃𝑤(𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → ∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
19	18	exgen 2002	. . 3 ⊢ ∃𝑧(¬ ∃𝑤(𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → ∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
20	19	19.37iv 1976	. 2 ⊢ (¬ ∃𝑤(𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
21	14, 20	pm2.61i 184	1 ⊢ ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 860  ∀wal 1566   = wceq 1568  ∃wex 1807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836

This theorem is referenced by:  prex  5409