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Mirrors > Home > ILE Home > Th. List > dtruarb | Unicode version |
Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). This theorem asserts the existence of two sets which do not equal each other; compare with dtruex 4541 in which we are given a set and go from there to a set which is not equal to it. (Contributed by Jim Kingdon, 2-Sep-2018.) |
Ref | Expression |
---|---|
dtruarb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | el 4162 | . . 3 | |
2 | ax-nul 4113 | . . . 4 | |
3 | sp 1504 | . . . 4 | |
4 | 2, 3 | eximii 1595 | . . 3 |
5 | eeanv 1925 | . . 3 | |
6 | 1, 4, 5 | mpbir2an 937 | . 2 |
7 | nelneq2 2272 | . . 3 | |
8 | 7 | 2eximi 1594 | . 2 |
9 | 6, 8 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wal 1346 wex 1485 |
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 609 ax-in2 610 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-13 2143 ax-14 2144 ax-ext 2152 ax-nul 4113 ax-pow 4158 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-cleq 2163 df-clel 2166 |
This theorem is referenced by: (None) |
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