Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > oprcl | Unicode version |
Description: If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
oprcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex2 2737 | . 2 | |
2 | df-op 3579 | . . . . . . 7 | |
3 | 2 | eleq2i 2231 | . . . . . 6 |
4 | df-clab 2151 | . . . . . 6 | |
5 | 3, 4 | bitri 183 | . . . . 5 |
6 | 3simpa 983 | . . . . . 6 | |
7 | 6 | sbimi 1751 | . . . . 5 |
8 | 5, 7 | sylbi 120 | . . . 4 |
9 | nfv 1515 | . . . . 5 | |
10 | 9 | sbf 1764 | . . . 4 |
11 | 8, 10 | sylib 121 | . . 3 |
12 | 11 | exlimiv 1585 | . 2 |
13 | 1, 12 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wex 1479 wsb 1749 wcel 2135 cab 2150 cvv 2721 csn 3570 cpr 3571 cop 3573 |
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-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-v 2723 df-op 3579 |
This theorem is referenced by: opth1 4208 opth 4209 0nelop 4220 |
Copyright terms: Public domain | W3C validator |