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Theorem uni0b 3605
 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

Proof of Theorem uni0b
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eq0 3239 . . . 4
21ralbii 2330 . . 3
3 ralcom4 2576 . . 3
42, 3bitri 173 . 2
5 dfss3 2935 . . 3
6 velsn 3392 . . . 4
76ralbii 2330 . . 3
85, 7bitri 173 . 2
9 eluni2 3584 . . . . 5
109notbii 594 . . . 4
1110albii 1359 . . 3
12 eq0 3239 . . 3
13 ralnex 2316 . . . 4
1413albii 1359 . . 3
1511, 12, 143bitr4i 201 . 2
164, 8, 153bitr4ri 202 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 98  wal 1241   wceq 1243   wcel 1393  wral 2306  wrex 2307   wss 2917  c0 3224  csn 3375  cuni 3580 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-uni 3581 This theorem is referenced by:  uni0c  3606  uni0  3607
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