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Mirrors > Home > ILE Home > Th. List > Mathboxes > ss1oel2o | Unicode version |
Description: Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4121 which more directly illustrates the contrast with el2oss1o 13188. (Contributed by Jim Kingdon, 8-Aug-2022.) |
Ref | Expression |
---|---|
ss1oel2o | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid01 4121 | . 2 EXMID | |
2 | df1o2 6326 | . . . . 5 | |
3 | 2 | sseq2i 3124 | . . . 4 |
4 | df2o2 6328 | . . . . . 6 | |
5 | 4 | eleq2i 2206 | . . . . 5 |
6 | vex 2689 | . . . . . 6 | |
7 | 6 | elpr 3548 | . . . . 5 |
8 | 5, 7 | bitri 183 | . . . 4 |
9 | 3, 8 | imbi12i 238 | . . 3 |
10 | 9 | albii 1446 | . 2 |
11 | 1, 10 | bitr4i 186 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wo 697 wal 1329 wceq 1331 wcel 1480 wss 3071 c0 3363 csn 3527 cpr 3528 EXMIDwem 4118 c1o 6306 c2o 6307 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-nul 4054 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-sn 3533 df-pr 3534 df-exmid 4119 df-suc 4293 df-1o 6313 df-2o 6314 |
This theorem is referenced by: (None) |
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