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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 4892 | . 2 ⊢ (𝐴 = if(0 ≤ 𝐴, 𝐴, 0) → (0 ≤ 𝐴 ↔ 0 ≤ if(0 ≤ 𝐴, 𝐴, 0))) | |
2 | breq2 4892 | . 2 ⊢ (0 = if(0 ≤ 𝐴, 𝐴, 0) → (0 ≤ 0 ↔ 0 ≤ if(0 ≤ 𝐴, 𝐴, 0))) | |
3 | 0re 10380 | . . 3 ⊢ 0 ∈ ℝ | |
4 | 3 | leidi 10911 | . 2 ⊢ 0 ≤ 0 |
5 | 1, 2, 4 | elimhyp 4370 | 1 ⊢ 0 ≤ if(0 ≤ 𝐴, 𝐴, 0) |
Colors of variables: wff setvar class |
Syntax hints: ifcif 4307 class class class wbr 4888 0cc0 10274 ≤ cle 10414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-addrcl 10335 ax-rnegex 10345 ax-cnre 10347 ax-pre-lttri 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 |
This theorem is referenced by: (None) |
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