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Theorem ncolcom 28545

Description: Swapping non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.)

**Hypotheses**
Ref	Expression
tglngval.p	⊢ 𝑃 = (Base‘𝐺)
tglngval.l	⊢ 𝐿 = (LineG‘𝐺)
tglngval.i	⊢ 𝐼 = (Itv‘𝐺)
tglngval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglngval.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tglngval.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
tgcolg.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
ncolrot	⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))

**Assertion**
Ref	Expression
ncolcom	⊢ (𝜑 → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))

**Proof of Theorem ncolcom**
Step	Hyp	Ref	Expression
1		ncolrot	. 2 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2		tglngval.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
3		tglngval.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
4		tglngval.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
5		tglngval.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝐺 ∈ TarskiG)
7		tglngval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝑌 ∈ 𝑃)
9		tglngval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
10	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝑋 ∈ 𝑃)
11		tgcolg.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝑍 ∈ 𝑃)
13		simpr 484	. . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
14	2, 3, 4, 6, 8, 10, 12, 13	colcom 28542	. 2 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
15	1, 14	mtand 815	1 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ‘cfv 6536 (class class class)co 7410 Basecbs 17233 TarskiGcstrkg 28411 Itvcitv 28417 LineGclng 28418

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pr 5407

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-iota 6489 df-fun 6538 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-trkgc 28432 df-trkgb 28433 df-trkgcb 28434 df-trkg 28437

This theorem is referenced by: ncolne2 28610 symquadlem 28673 midexlem 28676 outpasch 28739 acopyeu 28818 cgrg3col4 28837 tgasa1 28842

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