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Theorem tglnpt4 29123

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 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝐺 ∈ TarskiG)
6		tglnpt2.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝐴 ∈ ran 𝐿)
8		tglnpt4.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
9	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐴)
10	1, 2, 3, 5, 7, 9	tglnpt2 29121	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑋 ≠ 𝑧)
11		simplr 781	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑧 ∈ 𝐴)
12		simpr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑋 ≠ 𝑧)
13	12	neneqd 2961	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → ¬ 𝑋 = 𝑧)
14	4	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝐺 ∈ TarskiG)
15	6	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝐴 ∈ ran 𝐿)
16		tglnpt4.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
17	16	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝐵 ∈ ran 𝐿)
18		tglnpt4.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
19	18	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝐴 ≠ 𝐵)
20	8	ad4antr 745	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐴)
21		simp-4r 796	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐵)
22	20, 21	elind 4146	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ (𝐴 ∩ 𝐵))
23		simpllr 788	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐴)
24		simpr 490	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
25	23, 24	elind 4146	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (𝐴 ∩ 𝐵))
26	1, 2, 3, 14, 15, 17, 19, 22, 25	tglineineq 29111	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑋 = 𝑧)
27	13, 26	mtand 828	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → ¬ 𝑧 ∈ 𝐵)
28	11, 27	eldifd 3910	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑧 ∈ (𝐴 ∖ 𝐵))
29	12	necomd 3011	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑧 ≠ 𝑋)
30	28, 29	jca 521	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → (𝑧 ∈ (𝐴 ∖ 𝐵) ∧ 𝑧 ≠ 𝑋))
31	30	expl 463	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑧 ∈ 𝐴 ∧ 𝑋 ≠ 𝑧) → (𝑧 ∈ (𝐴 ∖ 𝐵) ∧ 𝑧 ≠ 𝑋)))
32	31	reximdv2 3173	. . 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 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐺 ∈ TarskiG)
35	6	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ ran 𝐿)
36	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑋 ∈ 𝐴)
37	1, 2, 3, 34, 35, 36	tglnpt2 29121	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) → ∃𝑧 ∈ 𝐴 𝑋 ≠ 𝑧)
38		simplr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑧 ∈ 𝐴)
39		simp-4r 796	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → (𝐴 ∩ 𝐵) = ∅)
40		simpllr 788	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐴)
41		simpr 490	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
42	40, 41	elind 4146	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (𝐴 ∩ 𝐵))
43	42	ne0d 4288	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → (𝐴 ∩ 𝐵) ≠ ∅)
44	43	neneqd 2961	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) ∧ 𝑧 ∈ 𝐵) → ¬ (𝐴 ∩ 𝐵) = ∅)
45	39, 44	pm2.65da 829	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → ¬ 𝑧 ∈ 𝐵)
46	38, 45	eldifd 3910	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑧 ∈ (𝐴 ∖ 𝐵))
47		simpr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑋 ≠ 𝑧)
48	47	necomd 3011	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → 𝑧 ≠ 𝑋)
49	46, 48	jca 521	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑧 ∈ 𝐴) ∧ 𝑋 ≠ 𝑧) → (𝑧 ∈ (𝐴 ∖ 𝐵) ∧ 𝑧 ≠ 𝑋))
50	49	expl 463	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑧 ∈ 𝐴 ∧ 𝑋 ≠ 𝑧) → (𝑧 ∈ (𝐴 ∖ 𝐵) ∧ 𝑧 ≠ 𝑋)))
51	50	reximdv2 3173	. . . . 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 728	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)
54		simpr 490	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) ≠ ∅) → (𝐴 ∩ 𝐵) ≠ ∅)
55	4	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝐺 ∈ TarskiG)
56	6	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝐴 ∈ ran 𝐿)
57	8	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑋 ∈ 𝐴)
58		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑦 ∈ (𝐴 ∩ 𝐵))
59	58	elin1d 4150	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑦 ∈ 𝐴)
60		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑦 ∈ (𝐴 ∩ 𝐵))
61	60	elin2d 4151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑦 ∈ 𝐵)
62	61	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑦 ∈ 𝐵)
63		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → ¬ 𝑋 ∈ 𝐵)
64		nelne2 3054	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝐵) → 𝑦 ≠ 𝑋)
65	62, 63, 64	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑦 ≠ 𝑋)
66	65	necomd 3011	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → 𝑋 ≠ 𝑦)
67	1, 2, 3, 55, 56, 57, 59, 66	tglnpt3 29122	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑋 ∧ 𝑧 ≠ 𝑦))
68		simpllr 788	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) → 𝑧 ∈ 𝐴)
69		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) → 𝑧 ≠ 𝑦)
70	69	neneqd 2961	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) → ¬ 𝑧 = 𝑦)
71	4	ad5antr 747	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝐺 ∈ TarskiG)
72	6	ad5antr 747	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝐴 ∈ ran 𝐿)
73	16	ad5antr 747	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝐵 ∈ ran 𝐿)
74	18	ad5antr 747	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝐴 ≠ 𝐵)
75	68	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐴)
76		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
77	75, 76	elind 4146	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (𝐴 ∩ 𝐵))
78	60	ad4antr 745	. . . . . . . . . . . . . 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 29111	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) ∧ 𝑧 ∈ 𝐵) → 𝑧 = 𝑦)
80	70, 79	mtand 828	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) → ¬ 𝑧 ∈ 𝐵)
81	68, 80	eldifd 3910	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) → 𝑧 ∈ (𝐴 ∖ 𝐵))
82		simplr 781	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) → 𝑧 ≠ 𝑋)
83	81, 82	jca 521	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≠ 𝑋) ∧ 𝑧 ≠ 𝑦) → (𝑧 ∈ (𝐴 ∖ 𝐵) ∧ 𝑧 ≠ 𝑋))
84	83	anasss 472	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴) ∧ (𝑧 ≠ 𝑋 ∧ 𝑧 ≠ 𝑦)) → (𝑧 ∈ (𝐴 ∖ 𝐵) ∧ 𝑧 ≠ 𝑋))
85	84	expl 463	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → ((𝑧 ∈ 𝐴 ∧ (𝑧 ≠ 𝑋 ∧ 𝑧 ≠ 𝑦)) → (𝑧 ∈ (𝐴 ∖ 𝐵) ∧ 𝑧 ≠ 𝑋)))
86	85	reximdv2 3173	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → (∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑋 ∧ 𝑧 ≠ 𝑦) → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋))
87	86	adantlr 728	. . . . . 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 728	. . . 4 ⊢ ((((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) ≠ ∅) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)
90	54, 89	n0limd 4301	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)
91	53, 90	pm2.61dane 3043	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝐵) → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)
92	33, 91	pm2.61dan 825	1 ⊢ (𝜑 → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∃wrex 3087 ∖ cdif 3896 ∩ cin 3898 ∅c0 4279 ran crn 5652 ‘cfv 6538 Basecbs 17387 TarskiGcstrkg 28889 Itvcitv 28895 LineGclng 28896

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-oadd 8480 df-er 8717 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-dju 9982 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-3 12406 df-n0 12607 df-xnn0 12680 df-z 12694 df-uz 12966 df-fz 13640 df-fzo 13789 df-hash 14475 df-word 14659 df-concat 14716 df-s1 14743 df-s2 14999 df-s3 15000 df-trkgc 28910 df-trkgb 28911 df-trkgcb 28912 df-trkg 28915 df-cgrg 28974

This theorem is used by: lnssplnglem 29269 perpeqlem 29347 perpprlng 29428 prlngex 29429 prlngmolem2 29431

W3C validator