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Mirrors > Home > ILE Home > Th. List > ordsucss | Unicode version |
Description: The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.) |
Ref | Expression |
---|---|
ordsucss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtr 4350 | . 2 | |
2 | trss 4083 | . . . . 5 | |
3 | snssi 3711 | . . . . . 6 | |
4 | 3 | a1i 9 | . . . . 5 |
5 | 2, 4 | jcad 305 | . . . 4 |
6 | unss 3291 | . . . 4 | |
7 | 5, 6 | syl6ib 160 | . . 3 |
8 | df-suc 4343 | . . . 4 | |
9 | 8 | sseq1i 3163 | . . 3 |
10 | 7, 9 | syl6ibr 161 | . 2 |
11 | 1, 10 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2135 cun 3109 wss 3111 csn 3570 wtr 4074 word 4334 csuc 4337 |
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-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 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-uni 3784 df-tr 4075 df-iord 4338 df-suc 4343 |
This theorem is referenced by: ordelsuc 4476 tfrlemibfn 6287 tfr1onlembfn 6303 tfrcllembfn 6316 sucinc2 6405 nndomo 6821 prarloclemn 7431 ennnfonelemhom 12291 ennnfonelemrn 12295 |
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