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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 3303 | . 2 ⊢ ∅ ⊆ {∅} | |
2 | uni0b 3652 | . 2 ⊢ (∪ ∅ = ∅ ↔ ∅ ⊆ {∅}) | |
3 | 1, 2 | mpbir 144 | 1 ⊢ ∪ ∅ = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1285 ⊆ wss 2984 ∅c0 3269 {csn 3422 ∪ cuni 3627 |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-dif 2986 df-in 2990 df-ss 2997 df-nul 3270 df-sn 3428 df-uni 3628 |
This theorem is referenced by: iununir 3785 unixp0im 4919 iotanul 4947 1st0 5848 2nd0 5849 brtpos0 5947 tpostpos 5959 nnsucuniel 6186 sup00 6603 |
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