Theorem preleqg 9583

Description: Equality of two unordered pairs when one member of each pair contains the other member. Closed form of preleq 9584. (Contributed by AV, 15-Jun-2022.)

Assertion

Ref	Expression
preleqg	⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem preleqg

Step	Hyp	Ref	Expression
1		elneq 9562	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵)
2	1	3ad2ant1 1149	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷) → 𝐴 ≠ 𝐵)
3		preq12nebg 4827	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
4	2, 3	syld3an3 1434	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
5		eleq12 2851	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 ∈ 𝐵 ↔ 𝐷 ∈ 𝐶))
6		elnotel 9578	. . . . . . . . . . 11 ⊢ (𝐷 ∈ 𝐶 → ¬ 𝐶 ∈ 𝐷)
7	6	pm2.21d 122	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝐶 → (𝐶 ∈ 𝐷 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
8	5, 7	biimtrdi 256	. . . . . . . . 9 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))))
9	8	com3l 90	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))))
10	9	a1d 26	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∈ 𝑉 → (𝐶 ∈ 𝐷 → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))))
11	10	3imp 1126	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷) → ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
12	11	com12 33	. . . . 5 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
13	12	jao1i 871	. . . 4 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
14	13	com12 33	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷) → (((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
15	4, 14	sylbid 243	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
16	15	imp 411	1 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  {cpr 4590

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-sep 5256  ax-pr 5404  ax-reg 9553

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-eprel 5561  df-fr 5614

This theorem is referenced by:  preleq  9584