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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 11240 | . . 3 ⊢ 0 ∈ ℝ | |
2 | 1 | ltnri 11347 | . 2 ⊢ ¬ 0 < 0 |
3 | c0ex 11232 | . . 3 ⊢ 0 ∈ V | |
4 | 3, 3 | gt-lth 48152 | . 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 5142 0cc0 11132 < clt 11272 > cgt 48146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-i2m1 11200 ax-rnegex 11203 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-ltxr 11277 df-gt 48148 |
This theorem is referenced by: (None) |
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