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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 3255 | . 2 ⊢ ∅ ⊆ {∅} | |
2 | uni0b 3605 | . 2 ⊢ (∪ ∅ = ∅ ↔ ∅ ⊆ {∅}) | |
3 | 1, 2 | mpbir 134 | 1 ⊢ ∪ ∅ = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ⊆ 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: iununir 3738 unixp0im 4854 iotanul 4882 1st0 5771 2nd0 5772 brtpos0 5867 tpostpos 5879 |
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