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Theorem axtgupdim2 25612
Description: Upper dimension axiom for dimension 2, Axiom A9 of [Schwabhauser] p. 13. Three points 𝑋, 𝑌 and 𝑍 equidistant to two given two points 𝑈 and 𝑉 must be colinear. (Contributed by Thierry Arnoux, 29-May-2019.) (Revised by Thierry Arnoux, 11-Jul-2020.)
Hypotheses
Ref Expression
axtrkge.p 𝑃 = (Base‘𝐺)
axtrkge.d = (dist‘𝐺)
axtrkge.i 𝐼 = (Itv‘𝐺)
axtgupdim2.x (𝜑𝑋𝑃)
axtgupdim2.y (𝜑𝑌𝑃)
axtgupdim2.z (𝜑𝑍𝑃)
axtgupdim2.u (𝜑𝑈𝑃)
axtgupdim2.v (𝜑𝑉𝑃)
axtgupdim2.0 (𝜑𝑈𝑉)
axtgupdim2.1 (𝜑 → (𝑈 𝑋) = (𝑉 𝑋))
axtgupdim2.2 (𝜑 → (𝑈 𝑌) = (𝑉 𝑌))
axtgupdim2.3 (𝜑 → (𝑈 𝑍) = (𝑉 𝑍))
axtgupdim2.w (𝜑𝐺𝑉)
axtgupdim2.g (𝜑 → ¬ 𝐺DimTarskiG≥3)
Assertion
Ref Expression
axtgupdim2 (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))

Proof of Theorem axtgupdim2
Dummy variables 𝑣 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axtgupdim2.1 . . . 4 (𝜑 → (𝑈 𝑋) = (𝑉 𝑋))
2 axtgupdim2.2 . . . 4 (𝜑 → (𝑈 𝑌) = (𝑉 𝑌))
3 axtgupdim2.3 . . . 4 (𝜑 → (𝑈 𝑍) = (𝑉 𝑍))
41, 2, 33jca 1149 . . 3 (𝜑 → ((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑍) = (𝑉 𝑍)))
5 axtgupdim2.0 . . . . . . 7 (𝜑𝑈𝑉)
6 axtgupdim2.g . . . . . . . . . . 11 (𝜑 → ¬ 𝐺DimTarskiG≥3)
7 axtgupdim2.w . . . . . . . . . . . . 13 (𝜑𝐺𝑉)
8 axtrkge.p . . . . . . . . . . . . . 14 𝑃 = (Base‘𝐺)
9 axtrkge.d . . . . . . . . . . . . . 14 = (dist‘𝐺)
10 axtrkge.i . . . . . . . . . . . . . 14 𝐼 = (Itv‘𝐺)
118, 9, 10istrkg3ld 25602 . . . . . . . . . . . . 13 (𝐺𝑉 → (𝐺DimTarskiG≥3 ↔ ∃𝑢𝑃𝑣𝑃 (𝑢𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
127, 11syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐺DimTarskiG≥3 ↔ ∃𝑢𝑃𝑣𝑃 (𝑢𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
1312notbid 308 . . . . . . . . . . 11 (𝜑 → (¬ 𝐺DimTarskiG≥3 ↔ ¬ ∃𝑢𝑃𝑣𝑃 (𝑢𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
146, 13mpbid 223 . . . . . . . . . 10 (𝜑 → ¬ ∃𝑢𝑃𝑣𝑃 (𝑢𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
15 ralnex2 3197 . . . . . . . . . 10 (∀𝑢𝑃𝑣𝑃 ¬ (𝑢𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ ¬ ∃𝑢𝑃𝑣𝑃 (𝑢𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
1614, 15sylibr 225 . . . . . . . . 9 (𝜑 → ∀𝑢𝑃𝑣𝑃 ¬ (𝑢𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
17 axtgupdim2.u . . . . . . . . . 10 (𝜑𝑈𝑃)
18 axtgupdim2.v . . . . . . . . . 10 (𝜑𝑉𝑃)
19 neeq1 3008 . . . . . . . . . . . . 13 (𝑢 = 𝑈 → (𝑢𝑣𝑈𝑣))
20 oveq1 6819 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑈 → (𝑢 𝑥) = (𝑈 𝑥))
2120eqeq1d 2776 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑈 → ((𝑢 𝑥) = (𝑣 𝑥) ↔ (𝑈 𝑥) = (𝑣 𝑥)))
22 oveq1 6819 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑈 → (𝑢 𝑦) = (𝑈 𝑦))
2322eqeq1d 2776 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑈 → ((𝑢 𝑦) = (𝑣 𝑦) ↔ (𝑈 𝑦) = (𝑣 𝑦)))
24 oveq1 6819 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑈 → (𝑢 𝑧) = (𝑈 𝑧))
2524eqeq1d 2776 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑈 → ((𝑢 𝑧) = (𝑣 𝑧) ↔ (𝑈 𝑧) = (𝑣 𝑧)))
2621, 23, 253anbi123d 1550 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑈 → (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ↔ ((𝑈 𝑥) = (𝑣 𝑥) ∧ (𝑈 𝑦) = (𝑣 𝑦) ∧ (𝑈 𝑧) = (𝑣 𝑧))))
2726anbi1d 616 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑈 → ((((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ (((𝑈 𝑥) = (𝑣 𝑥) ∧ (𝑈 𝑦) = (𝑣 𝑦) ∧ (𝑈 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
2827rexbidv 3204 . . . . . . . . . . . . . . 15 (𝑢 = 𝑈 → (∃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑧𝑃 (((𝑈 𝑥) = (𝑣 𝑥) ∧ (𝑈 𝑦) = (𝑣 𝑦) ∧ (𝑈 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
2928rexbidv 3204 . . . . . . . . . . . . . 14 (𝑢 = 𝑈 → (∃𝑦𝑃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑣 𝑥) ∧ (𝑈 𝑦) = (𝑣 𝑦) ∧ (𝑈 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
3029rexbidv 3204 . . . . . . . . . . . . 13 (𝑢 = 𝑈 → (∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑣 𝑥) ∧ (𝑈 𝑦) = (𝑣 𝑦) ∧ (𝑈 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
3119, 30anbi12d 617 . . . . . . . . . . . 12 (𝑢 = 𝑈 → ((𝑢𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ (𝑈𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑣 𝑥) ∧ (𝑈 𝑦) = (𝑣 𝑦) ∧ (𝑈 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
3231notbid 308 . . . . . . . . . . 11 (𝑢 = 𝑈 → (¬ (𝑢𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ ¬ (𝑈𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑣 𝑥) ∧ (𝑈 𝑦) = (𝑣 𝑦) ∧ (𝑈 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
33 neeq2 3009 . . . . . . . . . . . . 13 (𝑣 = 𝑉 → (𝑈𝑣𝑈𝑉))
34 oveq1 6819 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑉 → (𝑣 𝑥) = (𝑉 𝑥))
3534eqeq2d 2784 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑉 → ((𝑈 𝑥) = (𝑣 𝑥) ↔ (𝑈 𝑥) = (𝑉 𝑥)))
36 oveq1 6819 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑉 → (𝑣 𝑦) = (𝑉 𝑦))
3736eqeq2d 2784 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑉 → ((𝑈 𝑦) = (𝑣 𝑦) ↔ (𝑈 𝑦) = (𝑉 𝑦)))
38 oveq1 6819 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑉 → (𝑣 𝑧) = (𝑉 𝑧))
3938eqeq2d 2784 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑉 → ((𝑈 𝑧) = (𝑣 𝑧) ↔ (𝑈 𝑧) = (𝑉 𝑧)))
4035, 37, 393anbi123d 1550 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑉 → (((𝑈 𝑥) = (𝑣 𝑥) ∧ (𝑈 𝑦) = (𝑣 𝑦) ∧ (𝑈 𝑧) = (𝑣 𝑧)) ↔ ((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧))))
4140anbi1d 616 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑉 → ((((𝑈 𝑥) = (𝑣 𝑥) ∧ (𝑈 𝑦) = (𝑣 𝑦) ∧ (𝑈 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
4241rexbidv 3204 . . . . . . . . . . . . . . 15 (𝑣 = 𝑉 → (∃𝑧𝑃 (((𝑈 𝑥) = (𝑣 𝑥) ∧ (𝑈 𝑦) = (𝑣 𝑦) ∧ (𝑈 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑧𝑃 (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
4342rexbidv 3204 . . . . . . . . . . . . . 14 (𝑣 = 𝑉 → (∃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑣 𝑥) ∧ (𝑈 𝑦) = (𝑣 𝑦) ∧ (𝑈 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
4443rexbidv 3204 . . . . . . . . . . . . 13 (𝑣 = 𝑉 → (∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑣 𝑥) ∧ (𝑈 𝑦) = (𝑣 𝑦) ∧ (𝑈 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
4533, 44anbi12d 617 . . . . . . . . . . . 12 (𝑣 = 𝑉 → ((𝑈𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑣 𝑥) ∧ (𝑈 𝑦) = (𝑣 𝑦) ∧ (𝑈 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ (𝑈𝑉 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
4645notbid 308 . . . . . . . . . . 11 (𝑣 = 𝑉 → (¬ (𝑈𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑣 𝑥) ∧ (𝑈 𝑦) = (𝑣 𝑦) ∧ (𝑈 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ ¬ (𝑈𝑉 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
4732, 46rspc2v 3478 . . . . . . . . . 10 ((𝑈𝑃𝑉𝑃) → (∀𝑢𝑃𝑣𝑃 ¬ (𝑢𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) → ¬ (𝑈𝑉 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
4817, 18, 47syl2anc 574 . . . . . . . . 9 (𝜑 → (∀𝑢𝑃𝑣𝑃 ¬ (𝑢𝑣 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑢 𝑥) = (𝑣 𝑥) ∧ (𝑢 𝑦) = (𝑣 𝑦) ∧ (𝑢 𝑧) = (𝑣 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) → ¬ (𝑈𝑉 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
4916, 48mpd 15 . . . . . . . 8 (𝜑 → ¬ (𝑈𝑉 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
50 imnan 387 . . . . . . . 8 ((𝑈𝑉 → ¬ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ ¬ (𝑈𝑉 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
5149, 50sylibr 225 . . . . . . 7 (𝜑 → (𝑈𝑉 → ¬ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
525, 51mpd 15 . . . . . 6 (𝜑 → ¬ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))
53 ralnex3 3198 . . . . . 6 (∀𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ¬ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))
5452, 53sylibr 225 . . . . 5 (𝜑 → ∀𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))
55 axtgupdim2.x . . . . . 6 (𝜑𝑋𝑃)
56 axtgupdim2.y . . . . . 6 (𝜑𝑌𝑃)
57 axtgupdim2.z . . . . . 6 (𝜑𝑍𝑃)
58 oveq2 6820 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑈 𝑥) = (𝑈 𝑋))
59 oveq2 6820 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑉 𝑥) = (𝑉 𝑋))
6058, 59eqeq12d 2789 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝑈 𝑥) = (𝑉 𝑥) ↔ (𝑈 𝑋) = (𝑉 𝑋)))
61603anbi1d 1554 . . . . . . . . 9 (𝑥 = 𝑋 → (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ↔ ((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧))))
62 oveq1 6819 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥𝐼𝑦) = (𝑋𝐼𝑦))
6362eleq2d 2839 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑦)))
64 eleq1 2841 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑦)))
65 oveq1 6819 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
6665eleq2d 2839 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
6763, 64, 663orbi123d 1549 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))))
6867notbid 308 . . . . . . . . 9 (𝑥 = 𝑋 → (¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ ¬ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))))
6961, 68anbi12d 617 . . . . . . . 8 (𝑥 = 𝑋 → ((((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)))))
7069notbid 308 . . . . . . 7 (𝑥 = 𝑋 → (¬ (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ¬ (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)))))
71 oveq2 6820 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑈 𝑦) = (𝑈 𝑌))
72 oveq2 6820 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑉 𝑦) = (𝑉 𝑌))
7371, 72eqeq12d 2789 . . . . . . . . . 10 (𝑦 = 𝑌 → ((𝑈 𝑦) = (𝑉 𝑦) ↔ (𝑈 𝑌) = (𝑉 𝑌)))
74733anbi2d 1555 . . . . . . . . 9 (𝑦 = 𝑌 → (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ↔ ((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑧) = (𝑉 𝑧))))
75 oveq2 6820 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑋𝐼𝑦) = (𝑋𝐼𝑌))
7675eleq2d 2839 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑧 ∈ (𝑋𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑌)))
77 oveq2 6820 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑧𝐼𝑦) = (𝑧𝐼𝑌))
7877eleq2d 2839 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑋 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑌)))
79 eleq1 2841 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
8076, 78, 793orbi123d 1549 . . . . . . . . . 10 (𝑦 = 𝑌 → ((𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))))
8180notbid 308 . . . . . . . . 9 (𝑦 = 𝑌 → (¬ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)) ↔ ¬ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))))
8274, 81anbi12d 617 . . . . . . . 8 (𝑦 = 𝑌 → ((((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))) ↔ (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)))))
8382notbid 308 . . . . . . 7 (𝑦 = 𝑌 → (¬ (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))) ↔ ¬ (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)))))
84 oveq2 6820 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑈 𝑧) = (𝑈 𝑍))
85 oveq2 6820 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑉 𝑧) = (𝑉 𝑍))
8684, 85eqeq12d 2789 . . . . . . . . . 10 (𝑧 = 𝑍 → ((𝑈 𝑧) = (𝑉 𝑧) ↔ (𝑈 𝑍) = (𝑉 𝑍)))
87863anbi3d 1556 . . . . . . . . 9 (𝑧 = 𝑍 → (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ↔ ((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑍) = (𝑉 𝑍))))
88 eleq1 2841 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑧 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
89 oveq1 6819 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (𝑧𝐼𝑌) = (𝑍𝐼𝑌))
9089eleq2d 2839 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑋 ∈ (𝑧𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
91 oveq2 6820 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
9291eleq2d 2839 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑌 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
9388, 90, 923orbi123d 1549 . . . . . . . . . 10 (𝑧 = 𝑍 → ((𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
9493notbid 308 . . . . . . . . 9 (𝑧 = 𝑍 → (¬ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)) ↔ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
9587, 94anbi12d 617 . . . . . . . 8 (𝑧 = 𝑍 → ((((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))) ↔ (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑍) = (𝑉 𝑍)) ∧ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
9695notbid 308 . . . . . . 7 (𝑧 = 𝑍 → (¬ (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))) ↔ ¬ (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑍) = (𝑉 𝑍)) ∧ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
9770, 83, 96rspc3v 3481 . . . . . 6 ((𝑋𝑃𝑌𝑃𝑍𝑃) → (∀𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) → ¬ (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑍) = (𝑉 𝑍)) ∧ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
9855, 56, 57, 97syl3anc 1480 . . . . 5 (𝜑 → (∀𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (((𝑈 𝑥) = (𝑉 𝑥) ∧ (𝑈 𝑦) = (𝑉 𝑦) ∧ (𝑈 𝑧) = (𝑉 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) → ¬ (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑍) = (𝑉 𝑍)) ∧ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
9954, 98mpd 15 . . . 4 (𝜑 → ¬ (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑍) = (𝑉 𝑍)) ∧ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
100 imnan 387 . . . 4 ((((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑍) = (𝑉 𝑍)) → ¬ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) ↔ ¬ (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑍) = (𝑉 𝑍)) ∧ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
10199, 100sylibr 225 . . 3 (𝜑 → (((𝑈 𝑋) = (𝑉 𝑋) ∧ (𝑈 𝑌) = (𝑉 𝑌) ∧ (𝑈 𝑍) = (𝑉 𝑍)) → ¬ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
1024, 101mpd 15 . 2 (𝜑 → ¬ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
103102notnotrd 130 1 (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 383  w3o 1097  w3a 1098   = wceq 1634  wcel 2148  wne 2946  wral 3064  wrex 3065   class class class wbr 4797  cfv 6042  (class class class)co 6812  3c3 11294  Basecbs 16084  distcds 16178  DimTarskiGcstrkgld 25575  Itvcitv 25577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-cnex 10215  ax-resscn 10216  ax-1cn 10217  ax-icn 10218  ax-addcl 10219  ax-addrcl 10220  ax-mulcl 10221  ax-mulrcl 10222  ax-mulcom 10223  ax-addass 10224  ax-mulass 10225  ax-distr 10226  ax-i2m1 10227  ax-1ne0 10228  ax-1rid 10229  ax-rnegex 10230  ax-rrecex 10231  ax-cnre 10232  ax-pre-lttri 10233  ax-pre-lttrn 10234  ax-pre-ltadd 10235  ax-pre-mulgt0 10236
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-om 7234  df-1st 7336  df-2nd 7337  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-er 7917  df-en 8131  df-dom 8132  df-sdom 8133  df-pnf 10299  df-mnf 10300  df-xr 10301  df-ltxr 10302  df-le 10303  df-sub 10491  df-neg 10492  df-nn 11244  df-2 11302  df-3 11303  df-n0 11517  df-z 11602  df-uz 11911  df-fz 12556  df-fzo 12696  df-trkgld 25593
This theorem is referenced by: (None)
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