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Mirrors > Home > MPE Home > Th. List > elimge0 | Structured version Visualization version GIF version |
Description: Hypothesis for weak deduction theorem to eliminate 0 ≤ 𝐴. (Contributed by NM, 30-Jul-1999.) |
Ref | Expression |
---|---|
elimge0 | ⊢ 0 ≤ if(0 ≤ 𝐴, 𝐴, 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5113 | . 2 ⊢ (𝐴 = if(0 ≤ 𝐴, 𝐴, 0) → (0 ≤ 𝐴 ↔ 0 ≤ if(0 ≤ 𝐴, 𝐴, 0))) | |
2 | breq2 5113 | . 2 ⊢ (0 = if(0 ≤ 𝐴, 𝐴, 0) → (0 ≤ 0 ↔ 0 ≤ if(0 ≤ 𝐴, 𝐴, 0))) | |
3 | 0re 11165 | . . 3 ⊢ 0 ∈ ℝ | |
4 | 3 | leidi 11697 | . 2 ⊢ 0 ≤ 0 |
5 | 1, 2, 4 | elimhyp 4555 | 1 ⊢ 0 ≤ if(0 ≤ 𝐴, 𝐴, 0) |
Colors of variables: wff setvar class |
Syntax hints: ifcif 4490 class class class wbr 5109 0cc0 11059 ≤ cle 11198 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-resscn 11116 ax-1cn 11117 ax-addrcl 11120 ax-rnegex 11130 ax-cnre 11132 ax-pre-lttri 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 |
This theorem is referenced by: (None) |
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