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Mirrors > Home > MPE Home > Th. List > Mathboxes > ex-gt | Structured version Visualization version GIF version |
Description: Simple example of >, in this case, 0 is not greater than 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ex-gt | ⊢ ¬ 0 > 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10975 | . . 3 ⊢ 0 ∈ ℝ | |
2 | 1 | ltnri 11082 | . 2 ⊢ ¬ 0 < 0 |
3 | c0ex 10967 | . . 3 ⊢ 0 ∈ V | |
4 | 3, 3 | gt-lth 46396 | . 2 ⊢ (0 > 0 ↔ 0 < 0) |
5 | 2, 4 | mtbir 323 | 1 ⊢ ¬ 0 > 0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 class class class wbr 5076 0cc0 10869 < clt 11007 > cgt 46390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-i2m1 10937 ax-rnegex 10940 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5077 df-opab 5139 df-mpt 5160 df-id 5491 df-po 5505 df-so 5506 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-ltxr 11012 df-gt 46392 |
This theorem is referenced by: (None) |
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