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Theorem tglnpt4 28908

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 28906	. . 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 2970	. . . . . . . 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 28896	. . . . . . . 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 3020	. . . . . 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 3182	. . 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 28906	. . . . 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 2970	. . . . . . . . . 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 3020	. . . . . . . 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 3182	. . . . 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 3063	. . . . . . . . 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 3020	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑋 ≠ 𝑦)
67	1, 2, 3, 55, 56, 57, 59, 66	tglnpt3 28907	. . . . . 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 2970	. . . . . . . . . . . . 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 28896	. . . . . . . . . . . . 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 3182	. . . . . . 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 3052	. 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 1568 ∈ wcel 2150 ≠ wne 2965 ∃wrex 3096 ∖ cdif 3910 ∩ cin 3912 ∅c0 4294 ran crn 5666 ‘cfv 6540 Basecbs 17272 TarskiGcstrkg 28676 Itvcitv 28682 LineGclng 28683

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2152 ax-9 2160 ax-10 2183 ax-11 2199 ax-12 2220 ax-ext 2742 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5340 ax-pr 5408 ax-un 7736 ax-cnex 11159 ax-resscn 11160 ax-1cn 11161 ax-icn 11162 ax-addcl 11163 ax-addrcl 11164 ax-mulcl 11165 ax-mulrcl 11166 ax-mulcom 11167 ax-addass 11168 ax-mulass 11169 ax-distr 11170 ax-i2m1 11171 ax-1ne0 11172 ax-1rid 11173 ax-rnegex 11174 ax-rrecex 11175 ax-cnre 11176 ax-pre-lttri 11177 ax-pre-lttrn 11178 ax-pre-ltadd 11179 ax-pre-mulgt0 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2099 df-mo 2574 df-eu 2604 df-clab 2749 df-cleq 2762 df-clel 2845 df-nfc 2919 df-ne 2966 df-nel 3072 df-ral 3087 df-rex 3097 df-reu 3377 df-rab 3424 df-v 3464 df-sbc 3753 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 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5560 df-eprel 5565 df-po 5573 df-so 5574 df-fr 5618 df-we 5620 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-oadd 8460 df-er 8697 df-pm 8830 df-en 8947 df-dom 8948 df-sdom 8949 df-fin 8950 df-dju 9890 df-card 9928 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11446 df-neg 11447 df-nn 12237 df-2 12306 df-3 12307 df-n0 12508 df-xnn0 12581 df-z 12595 df-uz 12866 df-fz 13539 df-fzo 13686 df-hash 14370 df-word 14554 df-concat 14611 df-s1 14637 df-s2 14888 df-s3 14889 df-trkgc 28697 df-trkgb 28698 df-trkgcb 28699 df-trkg 28702 df-cgrg 28760

This theorem is referenced by: lnssplnglem 29051 perpeqlem 29127 perpprlng 29177 prlngex 29178 prlngmolem2 29180

W3C validator