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Mirrors > Home > ILE Home > Th. List > 0neqopab | Unicode version |
Description: The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) |
Ref | Expression |
---|---|
0neqopab |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . 2 | |
2 | elopab 4241 | . . 3 | |
3 | nfopab1 4056 | . . . . . 6 | |
4 | 3 | nfel2 2325 | . . . . 5 |
5 | 4 | nfn 1651 | . . . 4 |
6 | nfopab2 4057 | . . . . . . 7 | |
7 | 6 | nfel2 2325 | . . . . . 6 |
8 | 7 | nfn 1651 | . . . . 5 |
9 | vex 2733 | . . . . . . . 8 | |
10 | vex 2733 | . . . . . . . 8 | |
11 | 9, 10 | opnzi 4218 | . . . . . . 7 |
12 | nesym 2385 | . . . . . . . 8 | |
13 | pm2.21 612 | . . . . . . . 8 | |
14 | 12, 13 | sylbi 120 | . . . . . . 7 |
15 | 11, 14 | ax-mp 5 | . . . . . 6 |
16 | 15 | adantr 274 | . . . . 5 |
17 | 8, 16 | exlimi 1587 | . . . 4 |
18 | 5, 17 | exlimi 1587 | . . 3 |
19 | 2, 18 | sylbi 120 | . 2 |
20 | 1, 19 | pm2.65i 634 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1348 wex 1485 wcel 2141 wne 2340 c0 3414 cop 3584 copab 4047 |
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-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-opab 4049 |
This theorem is referenced by: (None) |
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