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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 5154 | . 2 ⊢ (𝐴 = if(0 ≤ 𝐴, 𝐴, 0) → (0 ≤ 𝐴 ↔ 0 ≤ if(0 ≤ 𝐴, 𝐴, 0))) | |
2 | breq2 5154 | . 2 ⊢ (0 = if(0 ≤ 𝐴, 𝐴, 0) → (0 ≤ 0 ↔ 0 ≤ if(0 ≤ 𝐴, 𝐴, 0))) | |
3 | 0re 11252 | . . 3 ⊢ 0 ∈ ℝ | |
4 | 3 | leidi 11784 | . 2 ⊢ 0 ≤ 0 |
5 | 1, 2, 4 | elimhyp 4595 | 1 ⊢ 0 ≤ if(0 ≤ 𝐴, 𝐴, 0) |
Colors of variables: wff setvar class |
Syntax hints: ifcif 4530 class class class wbr 5150 0cc0 11144 ≤ cle 11285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-resscn 11201 ax-1cn 11202 ax-addrcl 11205 ax-rnegex 11215 ax-cnre 11217 ax-pre-lttri 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 |
This theorem is referenced by: (None) |
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