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Theorem unineq 4024
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 2838 . . . . . . 7 ((𝐴𝐶) = (𝐵𝐶) → (𝑥 ∈ (𝐴𝐶) ↔ 𝑥 ∈ (𝐵𝐶)))
2 elin 3945 . . . . . . 7 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
3 elin 3945 . . . . . . 7 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
41, 2, 33bitr3g 302 . . . . . 6 ((𝐴𝐶) = (𝐵𝐶) → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
5 iba 511 . . . . . . 7 (𝑥𝐶 → (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐶)))
6 iba 511 . . . . . . 7 (𝑥𝐶 → (𝑥𝐵 ↔ (𝑥𝐵𝑥𝐶)))
75, 6bibi12d 334 . . . . . 6 (𝑥𝐶 → ((𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶))))
84, 7syl5ibr 236 . . . . 5 (𝑥𝐶 → ((𝐴𝐶) = (𝐵𝐶) → (𝑥𝐴𝑥𝐵)))
98adantld 474 . . . 4 (𝑥𝐶 → (((𝐴𝐶) = (𝐵𝐶) ∧ (𝐴𝐶) = (𝐵𝐶)) → (𝑥𝐴𝑥𝐵)))
10 uncom 3906 . . . . . . . . 9 (𝐴𝐶) = (𝐶𝐴)
11 uncom 3906 . . . . . . . . 9 (𝐵𝐶) = (𝐶𝐵)
1210, 11eqeq12i 2784 . . . . . . . 8 ((𝐴𝐶) = (𝐵𝐶) ↔ (𝐶𝐴) = (𝐶𝐵))
13 eleq2 2838 . . . . . . . 8 ((𝐶𝐴) = (𝐶𝐵) → (𝑥 ∈ (𝐶𝐴) ↔ 𝑥 ∈ (𝐶𝐵)))
1412, 13sylbi 207 . . . . . . 7 ((𝐴𝐶) = (𝐵𝐶) → (𝑥 ∈ (𝐶𝐴) ↔ 𝑥 ∈ (𝐶𝐵)))
15 elun 3902 . . . . . . 7 (𝑥 ∈ (𝐶𝐴) ↔ (𝑥𝐶𝑥𝐴))
16 elun 3902 . . . . . . 7 (𝑥 ∈ (𝐶𝐵) ↔ (𝑥𝐶𝑥𝐵))
1714, 15, 163bitr3g 302 . . . . . 6 ((𝐴𝐶) = (𝐵𝐶) → ((𝑥𝐶𝑥𝐴) ↔ (𝑥𝐶𝑥𝐵)))
18 biorf 896 . . . . . . 7 𝑥𝐶 → (𝑥𝐴 ↔ (𝑥𝐶𝑥𝐴)))
19 biorf 896 . . . . . . 7 𝑥𝐶 → (𝑥𝐵 ↔ (𝑥𝐶𝑥𝐵)))
2018, 19bibi12d 334 . . . . . 6 𝑥𝐶 → ((𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐶𝑥𝐴) ↔ (𝑥𝐶𝑥𝐵))))
2117, 20syl5ibr 236 . . . . 5 𝑥𝐶 → ((𝐴𝐶) = (𝐵𝐶) → (𝑥𝐴𝑥𝐵)))
2221adantrd 475 . . . 4 𝑥𝐶 → (((𝐴𝐶) = (𝐵𝐶) ∧ (𝐴𝐶) = (𝐵𝐶)) → (𝑥𝐴𝑥𝐵)))
239, 22pm2.61i 176 . . 3 (((𝐴𝐶) = (𝐵𝐶) ∧ (𝐴𝐶) = (𝐵𝐶)) → (𝑥𝐴𝑥𝐵))
2423eqrdv 2768 . 2 (((𝐴𝐶) = (𝐵𝐶) ∧ (𝐴𝐶) = (𝐵𝐶)) → 𝐴 = 𝐵)
25 uneq1 3909 . . 3 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
26 ineq1 3956 . . 3 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2725, 26jca 495 . 2 (𝐴 = 𝐵 → ((𝐴𝐶) = (𝐵𝐶) ∧ (𝐴𝐶) = (𝐵𝐶)))
2824, 27impbii 199 1 (((𝐴𝐶) = (𝐵𝐶) ∧ (𝐴𝐶) = (𝐵𝐶)) ↔ 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 826   = wceq 1630  wcel 2144  cun 3719  cin 3720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-un 3726  df-in 3728
This theorem is referenced by: (None)
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