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Mirrors > Home > MPE Home > Th. List > pleval2 | Structured version Visualization version Unicode version |
Description: Less-than-or-equal in terms of less-than. (sspss 3706 analog.) (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 8-Feb-2015.) |
Ref | Expression |
---|---|
pleval2.b | |
pleval2.l | |
pleval2.s |
Ref | Expression |
---|---|
pleval2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pleval2.b | . . . 4 | |
2 | pleval2.l | . . . 4 | |
3 | pleval2.s | . . . 4 | |
4 | 1, 2, 3 | pleval2i 16964 | . . 3 |
5 | 4 | 3adant1 1079 | . 2 |
6 | 2, 3 | pltle 16961 | . . 3 |
7 | 1, 2 | posref 16951 | . . . . 5 |
8 | 7 | 3adant3 1081 | . . . 4 |
9 | breq2 4657 | . . . 4 | |
10 | 8, 9 | syl5ibcom 235 | . . 3 |
11 | 6, 10 | jaod 395 | . 2 |
12 | 5, 11 | impbid 202 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wo 383 w3a 1037 wceq 1483 wcel 1990 class class class wbr 4653 cfv 5888 cbs 15857 cple 15948 cpo 16940 cplt 16941 |
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-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 |
This theorem is referenced by: pltletr 16971 plelttr 16972 tosso 17036 tlt3 29665 orngsqr 29804 |
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