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Mirrors > Home > MPE Home > Th. List > Mathboxes > ordtoplem | Structured version Visualization version Unicode version |
Description: Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.) |
Ref | Expression |
---|---|
ordtoplem.1 |
Ref | Expression |
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ordtoplem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2795 | . 2 | |
2 | ordeleqon 6988 | . . . . . 6 | |
3 | unon 7031 | . . . . . . . . 9 | |
4 | 3 | eqcomi 2631 | . . . . . . . 8 |
5 | id 22 | . . . . . . . 8 | |
6 | unieq 4444 | . . . . . . . 8 | |
7 | 4, 5, 6 | 3eqtr4a 2682 | . . . . . . 7 |
8 | 7 | orim2i 540 | . . . . . 6 |
9 | 2, 8 | sylbi 207 | . . . . 5 |
10 | 9 | orcomd 403 | . . . 4 |
11 | 10 | ord 392 | . . 3 |
12 | orduniorsuc 7030 | . . . 4 | |
13 | 12 | ord 392 | . . 3 |
14 | onuni 6993 | . . . 4 | |
15 | ordtoplem.1 | . . . 4 | |
16 | eleq1a 2696 | . . . 4 | |
17 | 14, 15, 16 | 3syl 18 | . . 3 |
18 | 11, 13, 17 | syl6c 70 | . 2 |
19 | 1, 18 | syl5bi 232 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wo 383 wceq 1483 wcel 1990 wne 2794 cuni 4436 word 5722 con0 5723 csuc 5725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-suc 5729 |
This theorem is referenced by: ordtop 32435 ordtopconn 32438 ordtopt0 32441 |
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