Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tlt3 | Structured version Visualization version Unicode version |
Description: In a Toset, two elements must compare strictly, or be equal. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
Ref | Expression |
---|---|
tlt3.b | |
tlt3.l |
Ref | Expression |
---|---|
tlt3 | Toset |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tlt3.b | . . . 4 | |
2 | eqid 2622 | . . . 4 | |
3 | tlt3.l | . . . 4 | |
4 | 1, 2, 3 | tlt2 29664 | . . 3 Toset |
5 | tospos 29658 | . . . . 5 Toset | |
6 | 1, 2, 3 | pleval2 16965 | . . . . . 6 |
7 | orcom 402 | . . . . . 6 | |
8 | 6, 7 | syl6bb 276 | . . . . 5 |
9 | 5, 8 | syl3an1 1359 | . . . 4 Toset |
10 | 9 | orbi1d 739 | . . 3 Toset |
11 | 4, 10 | mpbid 222 | . 2 Toset |
12 | df-3or 1038 | . 2 | |
13 | 11, 12 | sylibr 224 | 1 Toset |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wo 383 w3o 1036 w3a 1037 wceq 1483 wcel 1990 class class class wbr 4653 cfv 5888 cbs 15857 cple 15948 cpo 16940 cplt 16941 Tosetctos 17033 |
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-pow 4843 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-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-preset 16928 df-poset 16946 df-plt 16958 df-toset 17034 |
This theorem is referenced by: archirngz 29743 archiabllem1b 29746 archiabllem2b 29750 |
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