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Theorem unidif0 3911
Description: The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
Assertion
Ref Expression
unidif0 (A ∖ {∅}) = A

Proof of Theorem unidif0
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 3223 . . . . . . 7 (x y → ¬ y = ∅)
21pm4.71i 371 . . . . . 6 (x y ↔ (x y ¬ y = ∅))
32anbi1i 431 . . . . 5 ((x y y A) ↔ ((x y ¬ y = ∅) y A))
4 an32 496 . . . . 5 (((x y y A) ¬ y = ∅) ↔ ((x y ¬ y = ∅) y A))
5 anass 381 . . . . 5 (((x y y A) ¬ y = ∅) ↔ (x y (y A ¬ y = ∅)))
63, 4, 53bitr2ri 198 . . . 4 ((x y (y A ¬ y = ∅)) ↔ (x y y A))
76exbii 1493 . . 3 (y(x y (y A ¬ y = ∅)) ↔ y(x y y A))
8 eluni 3574 . . . 4 (x (A ∖ {∅}) ↔ y(x y y (A ∖ {∅})))
9 eldif 2921 . . . . . . 7 (y (A ∖ {∅}) ↔ (y A ¬ y {∅}))
10 elsn 3382 . . . . . . . . 9 (y {∅} ↔ y = ∅)
1110notbii 593 . . . . . . . 8 y {∅} ↔ ¬ y = ∅)
1211anbi2i 430 . . . . . . 7 ((y A ¬ y {∅}) ↔ (y A ¬ y = ∅))
139, 12bitri 173 . . . . . 6 (y (A ∖ {∅}) ↔ (y A ¬ y = ∅))
1413anbi2i 430 . . . . 5 ((x y y (A ∖ {∅})) ↔ (x y (y A ¬ y = ∅)))
1514exbii 1493 . . . 4 (y(x y y (A ∖ {∅})) ↔ y(x y (y A ¬ y = ∅)))
168, 15bitri 173 . . 3 (x (A ∖ {∅}) ↔ y(x y (y A ¬ y = ∅)))
17 eluni 3574 . . 3 (x Ay(x y y A))
187, 16, 173bitr4i 201 . 2 (x (A ∖ {∅}) ↔ x A)
1918eqriv 2034 1 (A ∖ {∅}) = A
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   = wceq 1242  wex 1378   wcel 1390  cdif 2908  c0 3218  {csn 3367   cuni 3571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-nul 3219  df-sn 3373  df-uni 3572
This theorem is referenced by: (None)
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