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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intss1 3752 |
. . 3
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2 | elirr 4416 |
. . . 4
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3 | ssel 3057 |
. . . 4
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4 | 2, 3 | mtoi 636 |
. . 3
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5 | 1, 4 | syl 14 |
. 2
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6 | 5 | adantl 273 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-setind 4412 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-v 2659 df-dif 3039 df-in 3043 df-ss 3050 df-sn 3499 df-int 3738 |
This theorem is referenced by: (None) |
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