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Mirrors > Home > MPE Home > Th. List > ordtri3or | Structured version Visualization version Unicode version |
Description: A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38. (Contributed by NM, 10-May-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
ordtri3or |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordin 5753 | . . . . . 6 | |
2 | ordirr 5741 | . . . . . 6 | |
3 | 1, 2 | syl 17 | . . . . 5 |
4 | ianor 509 | . . . . . 6 | |
5 | elin 3796 | . . . . . . 7 | |
6 | incom 3805 | . . . . . . . . 9 | |
7 | 6 | eleq1i 2692 | . . . . . . . 8 |
8 | 7 | anbi2i 730 | . . . . . . 7 |
9 | 5, 8 | bitri 264 | . . . . . 6 |
10 | 4, 9 | xchnxbir 323 | . . . . 5 |
11 | 3, 10 | sylib 208 | . . . 4 |
12 | inss1 3833 | . . . . . . . . . 10 | |
13 | ordsseleq 5752 | . . . . . . . . . 10 | |
14 | 12, 13 | mpbii 223 | . . . . . . . . 9 |
15 | 1, 14 | sylan 488 | . . . . . . . 8 |
16 | 15 | anabss1 855 | . . . . . . 7 |
17 | 16 | ord 392 | . . . . . 6 |
18 | df-ss 3588 | . . . . . 6 | |
19 | 17, 18 | syl6ibr 242 | . . . . 5 |
20 | ordin 5753 | . . . . . . . . 9 | |
21 | inss1 3833 | . . . . . . . . . 10 | |
22 | ordsseleq 5752 | . . . . . . . . . 10 | |
23 | 21, 22 | mpbii 223 | . . . . . . . . 9 |
24 | 20, 23 | sylan 488 | . . . . . . . 8 |
25 | 24 | anabss4 856 | . . . . . . 7 |
26 | 25 | ord 392 | . . . . . 6 |
27 | df-ss 3588 | . . . . . 6 | |
28 | 26, 27 | syl6ibr 242 | . . . . 5 |
29 | 19, 28 | orim12d 883 | . . . 4 |
30 | 11, 29 | mpd 15 | . . 3 |
31 | sspsstri 3709 | . . 3 | |
32 | 30, 31 | sylib 208 | . 2 |
33 | ordelpss 5751 | . . 3 | |
34 | biidd 252 | . . 3 | |
35 | ordelpss 5751 | . . . 4 | |
36 | 35 | ancoms 469 | . . 3 |
37 | 33, 34, 36 | 3orbi123d 1398 | . 2 |
38 | 32, 37 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 w3o 1036 wceq 1483 wcel 1990 cin 3573 wss 3574 wpss 3575 word 5722 |
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-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 |
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-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 |
This theorem is referenced by: ordtri1 5756 ordtri3OLD 5760 ordon 6982 ordeleqon 6988 smo11 7461 smoord 7462 omopth2 7664 r111 8638 tcrank 8747 domtriomlem 9264 axdc3lem2 9273 zorn2lem6 9323 grur1 9642 poseq 31750 soseq 31751 nosepon 31818 |
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