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Mirrors > Home > ILE Home > Th. List > onnmin | Unicode version |
Description: No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) (Constructive proof by Mario Carneiro and Jim Kingdon, 21-Jul-2019.) |
Ref | Expression |
---|---|
onnmin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intss1 3833 | . . 3 | |
2 | elirr 4512 | . . . 4 | |
3 | ssel 3131 | . . . 4 | |
4 | 2, 3 | mtoi 654 | . . 3 |
5 | 1, 4 | syl 14 | . 2 |
6 | 5 | adantl 275 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wcel 2135 wss 3111 cint 3818 con0 4335 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-v 2723 df-dif 3113 df-in 3117 df-ss 3124 df-sn 3576 df-int 3819 |
This theorem is referenced by: (None) |
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