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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 11739 | . 2 ⊢ 0 ≤ 0 | |
2 | c0ex 10635 | . . 3 ⊢ 0 ∈ V | |
3 | 2, 2 | gte-lteh 44845 | . 2 ⊢ (0 ≥ 0 ↔ 0 ≤ 0) |
4 | 1, 3 | mpbir 233 | 1 ⊢ 0 ≥ 0 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5066 0cc0 10537 ≤ cle 10676 ≥ cge-real 44839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-i2m1 10605 ax-rnegex 10608 ax-cnre 10610 ax-pre-lttri 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-gte 44841 |
This theorem is referenced by: (None) |
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