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Mirrors > Home > ILE Home > Th. List > uni0 | GIF version |
Description: The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.) |
Ref | Expression |
---|---|
uni0 | ⊢ ∪ ∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3282 | . 2 ⊢ ∅ ⊆ {∅} | |
2 | uni0b 3626 | . 2 ⊢ (∪ ∅ = ∅ ↔ ∅ ⊆ {∅}) | |
3 | 1, 2 | mpbir 144 | 1 ⊢ ∪ ∅ = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ⊆ wss 2973 ∅c0 3251 {csn 3398 ∪ cuni 3601 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-in 2979 df-ss 2986 df-nul 3252 df-sn 3404 df-uni 3602 |
This theorem is referenced by: iununir 3759 unixp0im 4874 iotanul 4902 1st0 5791 2nd0 5792 brtpos0 5890 tpostpos 5902 sup00 6416 |
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