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Theorem unineq 4239
Description: Infer equality from equalities of union and intersection. Exercise 20 of [Enderton] p. 32 and its converse. (Contributed by NM, 10-Aug-2004.)
Assertion
Ref Expression
unineq (((𝐴𝐶) = (𝐵𝐶) ∧ (𝐴𝐶) = (𝐵𝐶)) ↔ 𝐴 = 𝐵)

Proof of Theorem unineq
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2904 . . . . . . 7 ((𝐴𝐶) = (𝐵𝐶) → (𝑥 ∈ (𝐴𝐶) ↔ 𝑥 ∈ (𝐵𝐶)))
2 elin 3935 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
3 elin 3935 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
41, 2, 33bitr3g 316 . . . . . 6 ((𝐴𝐶) = (𝐵𝐶) → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
5 iba 531 . . . . . . 7 (𝑥𝐶 → (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐶)))
6 iba 531 . . . . . . 7 (𝑥𝐶 → (𝑥𝐵 ↔ (𝑥𝐵𝑥𝐶)))
75, 6bibi12d 349 . . . . . 6 (𝑥𝐶 → ((𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶))))
84, 7syl5ibr 249 . . . . 5 (𝑥𝐶 → ((𝐴𝐶) = (𝐵𝐶) → (𝑥𝐴𝑥𝐵)))
98adantld 494 . . . 4 (𝑥𝐶 → (((𝐴𝐶) = (𝐵𝐶) ∧ (𝐴𝐶) = (𝐵𝐶)) → (𝑥𝐴𝑥𝐵)))
10 uncom 4115 . . . . . . . . 9 (𝐴𝐶) = (𝐶𝐴)
11 uncom 4115 . . . . . . . . 9 (𝐵𝐶) = (𝐶𝐵)
1210, 11eqeq12i 2839 . . . . . . . 8 ((𝐴𝐶) = (𝐵𝐶) ↔ (𝐶𝐴) = (𝐶𝐵))
13 eleq2 2904 . . . . . . . 8 ((𝐶𝐴) = (𝐶𝐵) → (𝑥 ∈ (𝐶𝐴) ↔ 𝑥 ∈ (𝐶𝐵)))
1412, 13sylbi 220 . . . . . . 7 ((𝐴𝐶) = (𝐵𝐶) → (𝑥 ∈ (𝐶𝐴) ↔ 𝑥 ∈ (𝐶𝐵)))
15 elun 4111 . . . . . . 7 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶𝑥𝐴))
16 elun 4111 . . . . . . 7 (𝑥 ∈ (𝐶𝐵) ↔ (𝑥𝐶𝑥𝐵))
1714, 15, 163bitr3g 316 . . . . . 6 ((𝐴𝐶) = (𝐵𝐶) → ((𝑥𝐶𝑥𝐴) ↔ (𝑥𝐶𝑥𝐵)))
18 biorf 934 . . . . . . 7 𝑥𝐶 → (𝑥𝐴 ↔ (𝑥𝐶𝑥𝐴)))
19 biorf 934 . . . . . . 7 𝑥𝐶 → (𝑥𝐵 ↔ (𝑥𝐶𝑥𝐵)))
2018, 19bibi12d 349 . . . . . 6 𝑥𝐶 → ((𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐶𝑥𝐴) ↔ (𝑥𝐶𝑥𝐵))))
2117, 20syl5ibr 249 . . . . 5 𝑥𝐶 → ((𝐴𝐶) = (𝐵𝐶) → (𝑥𝐴𝑥𝐵)))
2221adantrd 495 . . . 4 𝑥𝐶 → (((𝐴𝐶) = (𝐵𝐶) ∧ (𝐴𝐶) = (𝐵𝐶)) → (𝑥𝐴𝑥𝐵)))
239, 22pm2.61i 185 . . 3 (((𝐴𝐶) = (𝐵𝐶) ∧ (𝐴𝐶) = (𝐵𝐶)) → (𝑥𝐴𝑥𝐵))
2423eqrdv 2822 . 2 (((𝐴𝐶) = (𝐵𝐶) ∧ (𝐴𝐶) = (𝐵𝐶)) → 𝐴 = 𝐵)
25 uneq1 4118 . . 3 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
26 ineq1 4166 . . 3 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2725, 26jca 515 . 2 (𝐴 = 𝐵 → ((𝐴𝐶) = (𝐵𝐶) ∧ (𝐴𝐶) = (𝐵𝐶)))
2824, 27impbii 212 1 (((𝐴𝐶) = (𝐵𝐶) ∧ (𝐴𝐶) = (𝐵𝐶)) ↔ 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2115  cun 3917  cin 3918
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rab 3142  df-v 3482  df-un 3924  df-in 3926
This theorem is referenced by: (None)
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