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Theorem uni0b 3596
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 3233 . . . 4 (x = ∅ ↔ y ¬ y x)
21ralbii 2324 . . 3 (x A x = ∅ ↔ x A y ¬ y x)
3 ralcom4 2570 . . 3 (x A y ¬ y xyx A ¬ y x)
42, 3bitri 173 . 2 (x A x = ∅ ↔ yx A ¬ y x)
5 dfss3 2929 . . 3 (A ⊆ {∅} ↔ x A x {∅})
6 elsn 3382 . . . 4 (x {∅} ↔ x = ∅)
76ralbii 2324 . . 3 (x A x {∅} ↔ x A x = ∅)
85, 7bitri 173 . 2 (A ⊆ {∅} ↔ x A x = ∅)
9 eluni2 3575 . . . . 5 (y Ax A y x)
109notbii 593 . . . 4 y A ↔ ¬ x A y x)
1110albii 1356 . . 3 (y ¬ y Ay ¬ x A y x)
12 eq0 3233 . . 3 ( A = ∅ ↔ y ¬ y A)
13 ralnex 2310 . . . 4 (x A ¬ y x ↔ ¬ x A y x)
1413albii 1356 . . 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 1240   = wceq 1242   wcel 1390  wral 2300  wrex 2301  wss 2911  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-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-uni 3572
This theorem is referenced by:  uni0c  3597  uni0  3598
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