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Theorem pr1eqbg 4857
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 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (𝐴 = 𝐶 ↔ {𝐴, 𝐵} = {𝐵, 𝐶}))

Proof of Theorem pr1eqbg
StepHypRef Expression
1 eqid 2737 . . . . 5 𝐵 = 𝐵
21biantru 529 . . . 4 (𝐴 = 𝐶 ↔ (𝐴 = 𝐶𝐵 = 𝐵))
32orbi2i 913 . . 3 (((𝐴 = 𝐵𝐵 = 𝐶) ∨ 𝐴 = 𝐶) ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵)))
43a1i 11 . 2 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (((𝐴 = 𝐵𝐵 = 𝐶) ∨ 𝐴 = 𝐶) ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
5 neneq 2946 . . . . 5 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
65adantl 481 . . . 4 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → ¬ 𝐴 = 𝐵)
76intnanrd 489 . . 3 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → ¬ (𝐴 = 𝐵𝐵 = 𝐶))
8 biorf 937 . . 3 (¬ (𝐴 = 𝐵𝐵 = 𝐶) → (𝐴 = 𝐶 ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ 𝐴 = 𝐶)))
97, 8syl 17 . 2 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (𝐴 = 𝐶 ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ 𝐴 = 𝐶)))
10 3simpa 1149 . . . . 5 ((𝐴𝑈𝐵𝑉𝐶𝑋) → (𝐴𝑈𝐵𝑉))
11 3simpc 1151 . . . . 5 ((𝐴𝑈𝐵𝑉𝐶𝑋) → (𝐵𝑉𝐶𝑋))
1210, 11jca 511 . . . 4 ((𝐴𝑈𝐵𝑉𝐶𝑋) → ((𝐴𝑈𝐵𝑉) ∧ (𝐵𝑉𝐶𝑋)))
1312adantr 480 . . 3 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → ((𝐴𝑈𝐵𝑉) ∧ (𝐵𝑉𝐶𝑋)))
14 preq12bg 4853 . . 3 (((𝐴𝑈𝐵𝑉) ∧ (𝐵𝑉𝐶𝑋)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
1513, 14syl 17 . 2 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
164, 9, 153bitr4d 311 1 (((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (𝐴 = 𝐶 ↔ {𝐴, 𝐵} = {𝐵, 𝐶}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629
This theorem is referenced by:  pr1nebg  4858
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