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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 4405 | . . . 4 | |
2 | 0elsucexmid.1 | . . . 4 | |
3 | suceq 4294 | . . . . . 6 | |
4 | 3 | eleq2d 2187 | . . . . 5 |
5 | 4 | rspcv 2759 | . . . 4 |
6 | 1, 2, 5 | mp2 16 | . . 3 |
7 | 0ex 4025 | . . . 4 | |
8 | 7 | elsuc 4298 | . . 3 |
9 | 6, 8 | mpbi 144 | . 2 |
10 | 7 | snid 3526 | . . . . 5 |
11 | biidd 171 | . . . . . 6 | |
12 | 11 | elrab3 2814 | . . . . 5 |
13 | 10, 12 | ax-mp 5 | . . . 4 |
14 | 13 | biimpi 119 | . . 3 |
15 | ordtriexmidlem2 4406 | . . . 4 | |
16 | 15 | eqcoms 2120 | . . 3 |
17 | 14, 16 | orim12i 733 | . 2 |
18 | 9, 17 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wb 104 wo 682 wceq 1316 wcel 1465 wral 2393 crab 2397 c0 3333 csn 3497 con0 4255 csuc 4257 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 |
This theorem is referenced by: (None) |
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