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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 4425 |
. 2
| |
| 2 | trss 4151 |
. . . . 5
| |
| 3 | snssi 3777 |
. . . . . 6
| |
| 4 | 3 | a1i 9 |
. . . . 5
|
| 5 | 2, 4 | jcad 307 |
. . . 4
|
| 6 | unss 3347 |
. . . 4
| |
| 7 | 5, 6 | imbitrdi 161 |
. . 3
|
| 8 | df-suc 4418 |
. . . 4
| |
| 9 | 8 | sseq1i 3219 |
. . 3
|
| 10 | 7, 9 | imbitrrdi 162 |
. 2
|
| 11 | 1, 10 | syl 14 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 |
| This theorem depends on definitions: df-bi 117 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-sn 3639 df-uni 3851 df-tr 4143 df-iord 4413 df-suc 4418 |
| This theorem is referenced by: ordelsuc 4553 tfrlemibfn 6414 tfr1onlembfn 6430 tfrcllembfn 6443 sucinc2 6532 nndomo 6961 prarloclemn 7612 ennnfonelemhom 12786 ennnfonelemrn 12790 |
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