Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 0elsucexmid | Unicode version |
Description: If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.) |
Ref | Expression |
---|---|
0elsucexmid.1 |
Ref | Expression |
---|---|
0elsucexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtriexmidlem 4501 | . . . 4 | |
2 | 0elsucexmid.1 | . . . 4 | |
3 | suceq 4385 | . . . . . 6 | |
4 | 3 | eleq2d 2240 | . . . . 5 |
5 | 4 | rspcv 2830 | . . . 4 |
6 | 1, 2, 5 | mp2 16 | . . 3 |
7 | 0ex 4114 | . . . 4 | |
8 | 7 | elsuc 4389 | . . 3 |
9 | 6, 8 | mpbi 144 | . 2 |
10 | 7 | snid 3612 | . . . . 5 |
11 | biidd 171 | . . . . . 6 | |
12 | 11 | elrab3 2887 | . . . . 5 |
13 | 10, 12 | ax-mp 5 | . . . 4 |
14 | 13 | biimpi 119 | . . 3 |
15 | ordtriexmidlem2 4502 | . . . 4 | |
16 | 15 | eqcoms 2173 | . . 3 |
17 | 14, 16 | orim12i 754 | . 2 |
18 | 9, 17 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wb 104 wo 703 wceq 1348 wcel 2141 wral 2448 crab 2452 c0 3414 csn 3581 con0 4346 csuc 4348 |
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-nul 4113 ax-pow 4158 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 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-uni 3795 df-tr 4086 df-iord 4349 df-on 4351 df-suc 4354 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |