leagne2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > leagne2		Structured version Visualization version GIF version

Theorem leagne2 28936

Description: Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.)

**Hypotheses**
Ref	Expression
isleag.p	⊢ 𝑃 = (Base‘𝐺)
isleag.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
isleag.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
isleag.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
isleag.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
isleag.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
isleag.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
isleag.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
leagne.1	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(≤_∠‘𝐺)⟨“𝐷𝐸𝐹”⟩)

**Assertion**
Ref	Expression
leagne2	⊢ (𝜑 → 𝐶 ≠ 𝐵)

Proof of Theorem leagne2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isleag.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2737	. . . 4 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
3		eqid 2737	. . . 4 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		isleag.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)) → 𝐺 ∈ TarskiG)
6		isleag.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)) → 𝐴 ∈ 𝑃)
8		isleag.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)) → 𝐵 ∈ 𝑃)
10		isleag.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)) → 𝐶 ∈ 𝑃)
12		isleag.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
13	12	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)) → 𝐷 ∈ 𝑃)
14		isleag.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
15	14	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)) → 𝐸 ∈ 𝑃)
16		simplr 769	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)) → 𝑥 ∈ 𝑃)
17		simprr 773	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)
18	1, 2, 3, 5, 7, 9, 11, 13, 15, 16, 17	cgrane2 28899	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)) → 𝐵 ≠ 𝐶)
19	18	necomd 2988	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)) → 𝐶 ≠ 𝐵)
20		leagne.1	. . 3 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(≤_∠‘𝐺)⟨“𝐷𝐸𝐹”⟩)
21		isleag.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
22	1, 4, 6, 8, 10, 12, 14, 21	isleag 28933	. . 3 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩(≤_∠‘𝐺)⟨“𝐷𝐸𝐹”⟩ ↔ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)))
23	20, 22	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩))
24	19, 23	r19.29a 3146	1 ⊢ (𝜑 → 𝐶 ≠ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 class class class wbr 5086 ‘cfv 6494 ⟨“cs3 14799 Basecbs 17174 TarskiGcstrkg 28513 Itvcitv 28519 hlGchlg 28686 cgrAccgra 28893 inAcinag 28921 ≤_∠cleag 28922

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-er 8638 df-map 8770 df-pm 8771 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-z 12520 df-uz 12784 df-fz 13457 df-fzo 13604 df-hash 14288 df-word 14471 df-concat 14528 df-s1 14554 df-s2 14805 df-s3 14806 df-trkgc 28534 df-trkgcb 28536 df-trkg 28539 df-cgrg 28597 df-hlg 28687 df-cgra 28894 df-leag 28932

This theorem is referenced by: (None)

W3C validator