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Theorem axtgupdim2 25561
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 1122 . . 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 25551 . . . . . . . . . . . . 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 307 . . . . . . . . . . 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 222 . . . . . . . . . 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 3175 . . . . . . . . . 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 224 . . . . . . . . 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 2986 . . . . . . . . . . . . 13 (𝑢 = 𝑈 → (𝑢𝑣𝑈𝑣))
20 oveq1 6812 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑈 → (𝑢 𝑥) = (𝑈 𝑥))
2120eqeq1d 2754 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑈 → ((𝑢 𝑥) = (𝑣 𝑥) ↔ (𝑈 𝑥) = (𝑣 𝑥)))
22 oveq1 6812 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑈 → (𝑢 𝑦) = (𝑈 𝑦))
2322eqeq1d 2754 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑈 → ((𝑢 𝑦) = (𝑣 𝑦) ↔ (𝑈 𝑦) = (𝑣 𝑦)))
24 oveq1 6812 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑈 → (𝑢 𝑧) = (𝑈 𝑧))
2524eqeq1d 2754 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑈 → ((𝑢 𝑧) = (𝑣 𝑧) ↔ (𝑈 𝑧) = (𝑣 𝑧)))
2621, 23, 253anbi123d 1540 . . . . . . . . . . . . . . . . 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 743 . . . . . . . . . . . . . . . 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 3182 . . . . . . . . . . . . . . 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 3182 . . . . . . . . . . . . . 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 3182 . . . . . . . . . . . . 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 749 . . . . . . . . . . . 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 307 . . . . . . . . . . 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 2987 . . . . . . . . . . . . 13 (𝑣 = 𝑉 → (𝑈𝑣𝑈𝑉))
34 oveq1 6812 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑉 → (𝑣 𝑥) = (𝑉 𝑥))
3534eqeq2d 2762 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑉 → ((𝑈 𝑥) = (𝑣 𝑥) ↔ (𝑈 𝑥) = (𝑉 𝑥)))
36 oveq1 6812 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑉 → (𝑣 𝑦) = (𝑉 𝑦))
3736eqeq2d 2762 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑉 → ((𝑈 𝑦) = (𝑣 𝑦) ↔ (𝑈 𝑦) = (𝑉 𝑦)))
38 oveq1 6812 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑉 → (𝑣 𝑧) = (𝑉 𝑧))
3938eqeq2d 2762 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑉 → ((𝑈 𝑧) = (𝑣 𝑧) ↔ (𝑈 𝑧) = (𝑉 𝑧)))
4035, 37, 393anbi123d 1540 . . . . . . . . . . . . . . . . 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 743 . . . . . . . . . . . . . . . 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 3182 . . . . . . . . . . . . . . 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 3182 . . . . . . . . . . . . . 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 3182 . . . . . . . . . . . . 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 749 . . . . . . . . . . . 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 307 . . . . . . . . . . 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 3453 . . . . . . . . . 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 696 . . . . . . . . 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 437 . . . . . . . 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 224 . . . . . . 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 3176 . . . . . 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 224 . . . . 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 6813 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑈 𝑥) = (𝑈 𝑋))
59 oveq2 6813 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑉 𝑥) = (𝑉 𝑋))
6058, 59eqeq12d 2767 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝑈 𝑥) = (𝑉 𝑥) ↔ (𝑈 𝑋) = (𝑉 𝑋)))
61603anbi1d 1544 . . . . . . . . 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 6812 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥𝐼𝑦) = (𝑋𝐼𝑦))
6362eleq2d 2817 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑦)))
64 eleq1 2819 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑦)))
65 oveq1 6812 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
6665eleq2d 2817 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
6763, 64, 663orbi123d 1539 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))))
6867notbid 307 . . . . . . . . 9 (𝑥 = 𝑋 → (¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ ¬ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧))))
6961, 68anbi12d 749 . . . . . . . 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 307 . . . . . . 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 6813 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑈 𝑦) = (𝑈 𝑌))
72 oveq2 6813 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑉 𝑦) = (𝑉 𝑌))
7371, 72eqeq12d 2767 . . . . . . . . . 10 (𝑦 = 𝑌 → ((𝑈 𝑦) = (𝑉 𝑦) ↔ (𝑈 𝑌) = (𝑉 𝑌)))
74733anbi2d 1545 . . . . . . . . 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 6813 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑋𝐼𝑦) = (𝑋𝐼𝑌))
7675eleq2d 2817 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑧 ∈ (𝑋𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑌)))
77 oveq2 6813 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑧𝐼𝑦) = (𝑧𝐼𝑌))
7877eleq2d 2817 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑋 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑌)))
79 eleq1 2819 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
8076, 78, 793orbi123d 1539 . . . . . . . . . 10 (𝑦 = 𝑌 → ((𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)) ↔ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))))
8180notbid 307 . . . . . . . . 9 (𝑦 = 𝑌 → (¬ (𝑧 ∈ (𝑋𝐼𝑦) ∨ 𝑋 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑋𝐼𝑧)) ↔ ¬ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))))
8274, 81anbi12d 749 . . . . . . . 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 307 . . . . . . 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 6813 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑈 𝑧) = (𝑈 𝑍))
85 oveq2 6813 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑉 𝑧) = (𝑉 𝑍))
8684, 85eqeq12d 2767 . . . . . . . . . 10 (𝑧 = 𝑍 → ((𝑈 𝑧) = (𝑉 𝑧) ↔ (𝑈 𝑍) = (𝑉 𝑍)))
87863anbi3d 1546 . . . . . . . . 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 2819 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑧 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
89 oveq1 6812 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (𝑧𝐼𝑌) = (𝑍𝐼𝑌))
9089eleq2d 2817 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑋 ∈ (𝑧𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
91 oveq2 6813 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
9291eleq2d 2817 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑌 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
9388, 90, 923orbi123d 1539 . . . . . . . . . 10 (𝑧 = 𝑍 → ((𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
9493notbid 307 . . . . . . . . 9 (𝑧 = 𝑍 → (¬ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)) ↔ ¬ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
9587, 94anbi12d 749 . . . . . . . 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 307 . . . . . . 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 3456 . . . . . 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 1473 . . . . 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 437 . . . 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 224 . . 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 128 1 (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3o 1071  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wral 3042  wrex 3043   class class class wbr 4796  cfv 6041  (class class class)co 6805  3c3 11255  Basecbs 16051  distcds 16144  DimTarskiGcstrkgld 25524  Itvcitv 25526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-3 11264  df-n0 11477  df-z 11562  df-uz 11872  df-fz 12512  df-fzo 12652  df-trkgld 25542
This theorem is referenced by: (None)
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