Theorem pr1eqbg 4787

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 2738	. . . . 5 ⊢ 𝐵 = 𝐵
2	1	biantru 530	. . . 4 ⊢ (𝐴 = 𝐶 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐵))
3	2	orbi2i 910	. . 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 2949	. . . . 5 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
6	5	adantl 482	. . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → ¬ 𝐴 = 𝐵)
7	6	intnanrd 490	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → ¬ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
8		biorf 934	. . 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 1147	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉))
11		3simpc 1149	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) → (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋))
12	10, 11	jca 512	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋)))
13	12	adantr 481	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋)))
14		preq12bg 4784	. . 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 311	1 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐶 ↔ {𝐴, 𝐵} = {𝐵, 𝐶}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-un 3892  df-sn 4562  df-pr 4564

This theorem is referenced by:  pr1nebg  4788