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Mathbox for David A. Wheeler |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ex-gte | Structured version Visualization version GIF version |
Description: Simple example of ≥, in this case, 0 is greater than or equal to 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-gte | ⊢ 0 ≥ 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0le0 12254 | . 2 ⊢ 0 ≤ 0 | |
2 | c0ex 11149 | . . 3 ⊢ 0 ∈ V | |
3 | 2, 2 | gte-lteh 47161 | . 2 ⊢ (0 ≥ 0 ↔ 0 ≤ 0) |
4 | 1, 3 | mpbir 230 | 1 ⊢ 0 ≥ 0 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5105 0cc0 11051 ≤ cle 11190 ≥ cge-real 47155 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-i2m1 11119 ax-rnegex 11122 ax-cnre 11124 ax-pre-lttri 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-gte 47157 |
This theorem is referenced by: (None) |
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