Theorem pr1eqbg 4618

Description: A (proper) pair is equal to another (maybe improper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)

Assertion

Ref	Expression
pr1eqbg	⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐶 ↔ {𝐴, 𝐵} = {𝐵, 𝐶}))

Proof of Theorem pr1eqbg

Step	Hyp	Ref	Expression
1		eqid 2777	. . . . 5 ⊢ 𝐵 = 𝐵
2	1	biantru 525	. . . 4 ⊢ (𝐴 = 𝐶 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐵))
3	2	orbi2i 899	. . 3 ⊢ (((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ 𝐴 = 𝐶) ↔ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐵)))
4	3	a1i 11	. 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → (((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ 𝐴 = 𝐶) ↔ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐵))))
5		neneq 2974	. . . . 5 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
6	5	adantl 475	. . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → ¬ 𝐴 = 𝐵)
7	6	intnanrd 485	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → ¬ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
8		biorf 923	. . 3 ⊢ (¬ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → (𝐴 = 𝐶 ↔ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ 𝐴 = 𝐶)))
9	7, 8	syl 17	. 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐶 ↔ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ 𝐴 = 𝐶)))
10		3simpa 1139	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
11		3simpc 1143	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) → (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋))
12	10, 11	jca 507	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋)))
13	12	adantr 474	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋)))
14		preq12bg 4614	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐵))))
15	13, 14	syl 17	. 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐵))))
16	4, 9, 15	3bitr4d 303	1 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐶 ↔ {𝐴, 𝐵} = {𝐵, 𝐶}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106   ≠ wne 2968  {cpr 4399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-v 3399  df-un 3796  df-sn 4398  df-pr 4400

This theorem is referenced by:  pr1nebg  4619