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Mirrors > Home > MPE Home > Th. List > inagne2 | Structured version Visualization version GIF version |
Description: Deduce inequality from the in-angle relation. (Contributed by Thierry Arnoux, 29-Oct-2021.) |
Ref | Expression |
---|---|
isinag.p | ⊢ 𝑃 = (Base‘𝐺) |
isinag.i | ⊢ 𝐼 = (Itv‘𝐺) |
isinag.k | ⊢ 𝐾 = (hlG‘𝐺) |
isinag.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
isinag.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
isinag.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
isinag.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
inagflat.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
inagswap.1 | ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐴𝐵𝐶”〉) |
Ref | Expression |
---|---|
inagne2 | ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inagswap.1 | . . . 4 ⊢ (𝜑 → 𝑋(inA‘𝐺)〈“𝐴𝐵𝐶”〉) | |
2 | isinag.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
3 | isinag.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | isinag.k | . . . . 5 ⊢ 𝐾 = (hlG‘𝐺) | |
5 | isinag.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
6 | isinag.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | isinag.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
8 | isinag.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
9 | inagflat.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | isinag 28558 | . . . 4 ⊢ (𝜑 → (𝑋(inA‘𝐺)〈“𝐴𝐵𝐶”〉 ↔ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋))))) |
11 | 1, 10 | mpbid 231 | . . 3 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵) ∧ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ (𝑥 = 𝐵 ∨ 𝑥(𝐾‘𝐵)𝑋)))) |
12 | 11 | simpld 494 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ 𝑋 ≠ 𝐵)) |
13 | 12 | simp2d 1140 | 1 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∃wrex 3062 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 〈“cs3 14790 Basecbs 17143 TarskiGcstrkg 28147 Itvcitv 28153 hlGchlg 28320 inAcinag 28555 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625 df-hash 14288 df-word 14462 df-concat 14518 df-s1 14543 df-s2 14796 df-s3 14797 df-inag 28557 |
This theorem is referenced by: leagne4 28572 |
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