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Theorem uni0b 3575
 Description: The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.)
Assertion
Ref Expression
uni0b ( A = ∅ ↔ A ⊆ {∅})

Proof of Theorem uni0b
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eq0 3212 . . . 4 (x = ∅ ↔ y ¬ y x)
21ralbii 2304 . . 3 (x A x = ∅ ↔ x A y ¬ y x)
3 ralcom4 2549 . . 3 (x A y ¬ y xyx A ¬ y x)
42, 3bitri 173 . 2 (x A x = ∅ ↔ yx A ¬ y x)
5 dfss3 2908 . . 3 (A ⊆ {∅} ↔ x A x {∅})
6 elsn 3361 . . . 4 (x {∅} ↔ x = ∅)
76ralbii 2304 . . 3 (x A x {∅} ↔ x A x = ∅)
85, 7bitri 173 . 2 (A ⊆ {∅} ↔ x A x = ∅)
9 eluni2 3554 . . . . 5 (y Ax A y x)
109notbii 581 . . . 4 y A ↔ ¬ x A y x)
1110albii 1335 . . 3 (y ¬ y Ay ¬ x A y x)
12 eq0 3212 . . 3 ( A = ∅ ↔ y ¬ y A)
13 ralnex 2290 . . . 4 (x A ¬ y x ↔ ¬ x A y x)
1413albii 1335 . . 3 (yx A ¬ y xy ¬ x A y x)
1511, 12, 143bitr4i 201 . 2 ( A = ∅ ↔ yx A ¬ y x)
164, 8, 153bitr4ri 202 1 ( A = ∅ ↔ A ⊆ {∅})
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 98  ∀wal 1224   = wceq 1226   ∈ wcel 1370  ∀wral 2280  ∃wrex 2281   ⊆ wss 2890  ∅c0 3197  {csn 3346  ∪ cuni 3550 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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-dif 2893  df-in 2897  df-ss 2904  df-nul 3198  df-sn 3352  df-uni 3551 This theorem is referenced by:  uni0c  3576  uni0  3577
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