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Theorem tglnpt4 28889

Description: Find a second point on a line, outside of a second line. (Contributed by Thierry Arnoux, 17-Jun-2026.)

**Hypotheses**
Ref	Expression
tglnpt2.p	⊢ 𝑃 = (Base‘𝐺)
tglnpt2.i	⊢ 𝐼 = (Itv‘𝐺)
tglnpt2.l	⊢ 𝐿 = (LineG‘𝐺)
tglnpt2.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglnpt2.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
tglnpt4.y	⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
tglnpt4.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
tglnpt4.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)

**Assertion**
Ref	Expression
tglnpt4	⊢ (𝜑 → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)

Distinct variable groups: 𝑧,𝐴 𝑧,𝐺 𝑧,𝐼 𝑧,𝑃 𝑧,𝑋 𝜑,𝑧 𝑧,𝐵 Allowed substitution hint: 𝐿(𝑧)

Proof of Theorem tglnpt4 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tglnpt2.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		tglnpt2.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
3		tglnpt2.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
4		tglnpt2.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝐺 ∈ TarskiG)
6		tglnpt2.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
7	6	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝐴 ∈ ran 𝐿)
8		tglnpt4.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
9	8	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐴)
10	1, 2, 3, 5, 7, 9	tglnpt2 28887	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑋 ≠ 𝑧)
11		simplr 780	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑧 ∈ 𝐴)
12		simpr 489	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑋 ≠ 𝑧)
13	12	neneqd 2969	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → ¬ 𝑋 = 𝑧)
14	4	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝐺 ∈ TarskiG)
15	6	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝐴 ∈ ran 𝐿)
16		tglnpt4.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
17	16	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝐵 ∈ ran 𝐿)
18		tglnpt4.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
19	18	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝐴 ≠ 𝐵)
20	8	ad4antr 744	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐴)
21		simp-4r 795	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐵)
22	20, 21	elind 4161	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ (𝐴 ∩ 𝐵))
23		simpllr 787	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐴)
24		simpr 489	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
25	23, 24	elind 4161	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (𝐴 ∩ 𝐵))
26	1, 2, 3, 14, 15, 17, 19, 22, 25	tglineineq 28877	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑋 = 𝑧)
27	13, 26	mtand 827	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → ¬ 𝑧 ∈ 𝐵)
28	11, 27	eldifd 3924	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑧 ∈ (𝐴 ∖ 𝐵))
29	12	necomd 3019	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑧 ≠ 𝑋)
30	28, 29	jca 520	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → (𝑧 ∈ (𝐴 ∖ 𝐵) ∧ 𝑧 ≠ 𝑋))
31	30	expl 462	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑧 ∈ 𝐴 ∧ 𝑋 ≠ 𝑧) → (𝑧 ∈ (𝐴 ∖ 𝐵) ∧ 𝑧 ≠ 𝑋)))
32	31	reximdv2 3181	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (∃𝑧 ∈ 𝐴 𝑋 ≠ 𝑧 → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋))
33	10, 32	mpd 16	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)
34	4	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐺 ∈ TarskiG)
35	6	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ ran 𝐿)
36	8	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑋 ∈ 𝐴)
37	1, 2, 3, 34, 35, 36	tglnpt2 28887	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) → ∃𝑧 ∈ 𝐴 𝑋 ≠ 𝑧)
38		simplr 780	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑧 ∈ 𝐴)
39		simp-4r 795	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → (𝐴 ∩ 𝐵) = ∅)
40		simpllr 787	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐴)
41		simpr 489	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
42	40, 41	elind 4161	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (𝐴 ∩ 𝐵))
43	42	ne0d 4303	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → (𝐴 ∩ 𝐵) ≠ ∅)
44	43	neneqd 2969	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → ¬ (𝐴 ∩ 𝐵) = ∅)
45	39, 44	pm2.65da 828	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → ¬ 𝑧 ∈ 𝐵)
46	38, 45	eldifd 3924	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑧 ∈ (𝐴 ∖ 𝐵))
47		simpr 489	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑋 ≠ 𝑧)
48	47	necomd 3019	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑧 ≠ 𝑋)
49	46, 48	jca 520	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → (𝑧 ∈ (𝐴 ∖ 𝐵) ∧ 𝑧 ≠ 𝑋))
50	49	expl 462	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑧 ∈ 𝐴 ∧ 𝑋 ≠ 𝑧) → (𝑧 ∈ (𝐴 ∖ 𝐵) ∧ 𝑧 ≠ 𝑋)))
51	50	reximdv2 3181	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) → (∃𝑧 ∈ 𝐴 𝑋 ≠ 𝑧 → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋))
52	37, 51	mpd 16	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)
53	52	adantlr 727	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)
54		simpr 489	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (𝐴 ∩ 𝐵) ≠ ∅)
55	4	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝐺 ∈ TarskiG)
56	6	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝐴 ∈ ran 𝐿)
57	8	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑋 ∈ 𝐴)
58		simpr 489	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑦 ∈ (𝐴 ∩ 𝐵))
59	58	elin1d 4165	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑦 ∈ 𝐴)
60		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑦 ∈ (𝐴 ∩ 𝐵))
61	60	elin2d 4166	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑦 ∈ 𝐵)
62	61	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑦 ∈ 𝐵)
63		simplr 780	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → ¬ 𝑋 ∈ 𝐵)
64		nelne2 3062	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝐵) → 𝑦 ≠ 𝑋)
65	62, 63, 64	syl2anc 595	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑦 ≠ 𝑋)
66	65	necomd 3019	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑋 ≠ 𝑦)
67	1, 2, 3, 55, 56, 57, 59, 66	tglnpt3 28888	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑋 ∧ 𝑧 ≠ 𝑦))
68		simpllr 787	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) → 𝑧 ∈ 𝐴)
69		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) → 𝑧 ≠ 𝑦)
70	69	neneqd 2969	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) → ¬ 𝑧 = 𝑦)
71	4	ad5antr 746	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝐺 ∈ TarskiG)
72	6	ad5antr 746	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝐴 ∈ ran 𝐿)
73	16	ad5antr 746	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝐵 ∈ ran 𝐿)
74	18	ad5antr 746	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝐴 ≠ 𝐵)
75	68	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐴)
76		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
77	75, 76	elind 4161	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (𝐴 ∩ 𝐵))
78	60	ad4antr 744	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝑦 ∈ (𝐴 ∩ 𝐵))
79	1, 2, 3, 71, 72, 73, 74, 77, 78	tglineineq 28877	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝑧 = 𝑦)
80	70, 79	mtand 827	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) → ¬ 𝑧 ∈ 𝐵)
81	68, 80	eldifd 3924	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) → 𝑧 ∈ (𝐴 ∖ 𝐵))
82		simplr 780	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) → 𝑧 ≠ 𝑋)
83	81, 82	jca 520	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) → (𝑧 ∈ (𝐴 ∖ 𝐵) ∧ 𝑧 ≠ 𝑋))
84	83	anasss 471	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑧 ≠ 𝑋 ∧ 𝑧 ≠ 𝑦)) → (𝑧 ∈ (𝐴 ∖ 𝐵) ∧ 𝑧 ≠ 𝑋))
85	84	expl 462	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → ((𝑧 ∈ 𝐴 ∧ (𝑧 ≠ 𝑋 ∧ 𝑧 ≠ 𝑦)) → (𝑧 ∈ (𝐴 ∖ 𝐵) ∧ 𝑧 ≠ 𝑋)))
86	85	reximdv2 3181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → (∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑋 ∧ 𝑧 ≠ 𝑦) → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋))
87	86	adantlr 727	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → (∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑋 ∧ 𝑧 ≠ 𝑦) → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋))
88	67, 87	mpd 16	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)
89	88	adantlr 727	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) ≠ ∅) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)
90	54, 89	n0limd 4316	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)
91	53, 90	pm2.61dane 3051	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)
92	33, 91	pm2.61dan 824	1 ⊢ (𝜑 → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∃wrex 3095 ∖ cdif 3910 ∩ cin 3912 ∅c0 4294 ran crn 5663 ‘cfv 6537 Basecbs 17268 TarskiGcstrkg 28661 Itvcitv 28667 LineGclng 28668

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-oadd 8456 df-er 8693 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-dju 9886 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-n0 12504 df-xnn0 12577 df-z 12591 df-uz 12862 df-fz 13535 df-fzo 13682 df-hash 14366 df-word 14550 df-concat 14607 df-s1 14633 df-s2 14884 df-s3 14885 df-trkgc 28682 df-trkgb 28683 df-trkgcb 28684 df-trkg 28687 df-cgrg 28745

This theorem is referenced by: lnssplnglem 29030 perpeqlem 29104 perpprlng 29152 prlngex 29153 prlngmolem2 29155

W3C validator