Theorem traxext 45285

Description: A transitive class models the Axiom of Extensionality ax-ext 2709. Lemma II.2.4(1) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 11-Sep-2025.)

Assertion

Ref	Expression
traxext	⊢ (Tr 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝑦,𝑧,𝑀

Proof of Theorem traxext

Step	Hyp	Ref	Expression
1		df-ral 3053	. . . 4 ⊢ (∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑀 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
2		trel 5214	. . . . . . . . . . 11 ⊢ (Tr 𝑀 → ((𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑀) → 𝑧 ∈ 𝑀))
3	2	ancomsd 465	. . . . . . . . . 10 ⊢ (Tr 𝑀 → ((𝑥 ∈ 𝑀 ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑀))
4	3	expdimp 452	. . . . . . . . 9 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑀))
5	4	adantrr 718	. . . . . . . 8 ⊢ ((Tr 𝑀 ∧ (𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀)) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑀))
6	5	adantr 480	. . . . . . 7 ⊢ (((Tr 𝑀 ∧ (𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀)) ∧ (𝑧 ∈ 𝑀 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑀))
7		trel 5214	. . . . . . . . . . 11 ⊢ (Tr 𝑀 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑀) → 𝑧 ∈ 𝑀))
8	7	ancomsd 465	. . . . . . . . . 10 ⊢ (Tr 𝑀 → ((𝑦 ∈ 𝑀 ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ 𝑀))
9	8	expdimp 452	. . . . . . . . 9 ⊢ ((Tr 𝑀 ∧ 𝑦 ∈ 𝑀) → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑀))
10	9	adantrl 717	. . . . . . . 8 ⊢ ((Tr 𝑀 ∧ (𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀)) → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑀))
11	10	adantr 480	. . . . . . 7 ⊢ (((Tr 𝑀 ∧ (𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀)) ∧ (𝑧 ∈ 𝑀 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))) → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑀))
12		simpr 484	. . . . . . 7 ⊢ (((Tr 𝑀 ∧ (𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀)) ∧ (𝑧 ∈ 𝑀 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))) → (𝑧 ∈ 𝑀 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
13	6, 11, 12	pm5.21ndd 379	. . . . . 6 ⊢ (((Tr 𝑀 ∧ (𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀)) ∧ (𝑧 ∈ 𝑀 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))) → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
14	13	ex 412	. . . . 5 ⊢ ((Tr 𝑀 ∧ (𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀)) → ((𝑧 ∈ 𝑀 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
15	14	alimdv 1918	. . . 4 ⊢ ((Tr 𝑀 ∧ (𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀)) → (∀𝑧(𝑧 ∈ 𝑀 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) → ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
16	1, 15	biimtrid 242	. . 3 ⊢ ((Tr 𝑀 ∧ (𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀)) → (∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
17		ax-ext 2709	. . 3 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
18	16, 17	syl6 35	. 2 ⊢ ((Tr 𝑀 ∧ (𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀)) → (∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))
19	18	ralrimivva 3180	1 ⊢ (Tr 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   ∈ wcel 2114  ∀wral 3052  Tr wtr 5206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3443  df-ss 3919  df-uni 4865  df-tr 5207

This theorem is referenced by:  wfaxext  45301