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Mirrors > Home > ILE Home > Th. List > uni0 | Unicode 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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3306 |
. 2
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2 | uni0b 3655 |
. 2
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3 | 1, 2 | mpbir 144 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() |
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 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 |
This theorem depends on definitions: df-bi 115 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ral 2360 df-rex 2361 df-v 2616 df-dif 2988 df-in 2992 df-ss 2999 df-nul 3273 df-sn 3431 df-uni 3631 |
This theorem is referenced by: iununir 3788 unixp0im 4924 iotanul 4952 1st0 5853 2nd0 5854 brtpos0 5952 tpostpos 5964 nnsucuniel 6191 sup00 6619 nnpredcl 11246 |
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