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Theorem iununi 4050
 Description: A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iununi ((B = A = ) ↔ (AB) = x B (Ax))
Distinct variable groups:   x,A   x,B

Proof of Theorem iununi
StepHypRef Expression
1 df-ne 2518 . . . . . . 7 (B ↔ ¬ B = )
2 iunconst 3977 . . . . . . 7 (Bx B A = A)
31, 2sylbir 204 . . . . . 6 B = x B A = A)
4 iun0 4022 . . . . . . 7 x B =
5 id 19 . . . . . . . 8 (A = A = )
65iuneq2d 3994 . . . . . . 7 (A = x B A = x B )
74, 6, 53eqtr4a 2411 . . . . . 6 (A = x B A = A)
83, 7ja 153 . . . . 5 ((B = A = ) → x B A = A)
98eqcomd 2358 . . . 4 ((B = A = ) → A = x B A)
109uneq1d 3417 . . 3 ((B = A = ) → (Ax B x) = (x B Ax B x))
11 uniiun 4019 . . . 4 B = x B x
1211uneq2i 3415 . . 3 (AB) = (Ax B x)
13 iunun 4046 . . 3 x B (Ax) = (x B Ax B x)
1410, 12, 133eqtr4g 2410 . 2 ((B = A = ) → (AB) = x B (Ax))
15 unieq 3900 . . . . . . 7 (B = B = )
16 uni0 3918 . . . . . . 7 =
1715, 16syl6eq 2401 . . . . . 6 (B = B = )
1817uneq2d 3418 . . . . 5 (B = → (AB) = (A))
19 un0 3575 . . . . 5 (A) = A
2018, 19syl6eq 2401 . . . 4 (B = → (AB) = A)
21 iuneq1 3982 . . . . 5 (B = x B (Ax) = x (Ax))
22 0iun 4023 . . . . 5 x (Ax) =
2321, 22syl6eq 2401 . . . 4 (B = x B (Ax) = )
2420, 23eqeq12d 2367 . . 3 (B = → ((AB) = x B (Ax) ↔ A = ))
2524biimpcd 215 . 2 ((AB) = x B (Ax) → (B = A = ))
2614, 25impbii 180 1 ((B = A = ) ↔ (AB) = x B (Ax))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   = wceq 1642   ≠ wne 2516   ∪ cun 3207  ∅c0 3550  ∪cuni 3891  ∪ciun 3969 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-uni 3892  df-iun 3971 This theorem is referenced by: (None)
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