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Theorem leagne3 28907

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
leagne3	⊢ (𝜑 → 𝐷 ≠ 𝐸)

Proof of Theorem leagne3 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isleag.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2735	. . . 4 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
3		eqid 2735	. . . 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	cgrane3 28870	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)) → 𝐸 ≠ 𝐷)
19	18	necomd 2985	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩ ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑥”⟩)) → 𝐷 ≠ 𝐸)
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 28903	. . 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 3143	1 ⊢ (𝜑 → 𝐷 ≠ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2930 ∃wrex 3059 class class class wbr 5074 ‘cfv 6487 ⟨“cs3 14793 Basecbs 17168 TarskiGcstrkg 28483 Itvcitv 28489 hlGchlg 28656 cgrAccgra 28863 inAcinag 28891 ≤_∠cleag 28892

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 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8632 df-map 8764 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-3 12234 df-n0 12427 df-z 12514 df-uz 12778 df-fz 13451 df-fzo 13598 df-hash 14282 df-word 14465 df-concat 14522 df-s1 14548 df-s2 14799 df-s3 14800 df-hlg 28657 df-cgra 28864 df-leag 28902

This theorem is referenced by: (None)

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