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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 3396 | . 2 ⊢ ∅ ⊆ {∅} | |
2 | uni0b 3756 | . 2 ⊢ (∪ ∅ = ∅ ↔ ∅ ⊆ {∅}) | |
3 | 1, 2 | mpbir 145 | 1 ⊢ ∪ ∅ = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ⊆ wss 3066 ∅c0 3358 {csn 3522 ∪ cuni 3731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-in 3072 df-ss 3079 df-nul 3359 df-sn 3528 df-uni 3732 |
This theorem is referenced by: iununir 3891 nnpredcl 4531 unixp0im 5070 iotanul 5098 1st0 6035 2nd0 6036 brtpos0 6142 tpostpos 6154 nnsucuniel 6384 sup00 6883 0opn 12162 |
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