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Theorem axtgupdim2 25270
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 1240 . . 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 25260 . . . . . . . . . . . . 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 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 3038 . . . . . . . . . 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 2852 . . . . . . . . . . . . 13 (𝑢 = 𝑈 → (𝑢𝑣𝑈𝑣))
20 oveq1 6611 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑈 → (𝑢 𝑥) = (𝑈 𝑥))
2120eqeq1d 2623 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑈 → ((𝑢 𝑥) = (𝑣 𝑥) ↔ (𝑈 𝑥) = (𝑣 𝑥)))
22 oveq1 6611 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑈 → (𝑢 𝑦) = (𝑈 𝑦))
2322eqeq1d 2623 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑈 → ((𝑢 𝑦) = (𝑣 𝑦) ↔ (𝑈 𝑦) = (𝑣 𝑦)))
24 oveq1 6611 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑈 → (𝑢 𝑧) = (𝑈 𝑧))
2524eqeq1d 2623 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑈 → ((𝑢 𝑧) = (𝑣 𝑧) ↔ (𝑈 𝑧) = (𝑣 𝑧)))
2621, 23, 253anbi123d 1396 . . . . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . . . 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 3045 . . . . . . . . . . . . . . 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 3045 . . . . . . . . . . . . . 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 3045 . . . . . . . . . . . . 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 746 . . . . . . . . . . . 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 2853 . . . . . . . . . . . . 13 (𝑣 = 𝑉 → (𝑈𝑣𝑈𝑉))
34 oveq1 6611 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑉 → (𝑣 𝑥) = (𝑉 𝑥))
3534eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑉 → ((𝑈 𝑥) = (𝑣 𝑥) ↔ (𝑈 𝑥) = (𝑉 𝑥)))
36 oveq1 6611 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑉 → (𝑣 𝑦) = (𝑉 𝑦))
3736eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑉 → ((𝑈 𝑦) = (𝑣 𝑦) ↔ (𝑈 𝑦) = (𝑉 𝑦)))
38 oveq1 6611 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑉 → (𝑣 𝑧) = (𝑉 𝑧))
3938eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑉 → ((𝑈 𝑧) = (𝑣 𝑧) ↔ (𝑈 𝑧) = (𝑉 𝑧)))
4035, 37, 393anbi123d 1396 . . . . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . . . 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 3045 . . . . . . . . . . . . . . 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 3045 . . . . . . . . . . . . . 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 3045 . . . . . . . . . . . . 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 746 . . . . . . . . . . . 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 3306 . . . . . . . . . 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 692 . . . . . . . . 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 438 . . . . . . . 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 3039 . . . . . 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 6612 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑈 𝑥) = (𝑈 𝑋))
59 oveq2 6612 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑉 𝑥) = (𝑉 𝑋))
6058, 59eqeq12d 2636 . . . . . . . . . 10 (𝑥 = 𝑋 → ((𝑈 𝑥) = (𝑉 𝑥) ↔ (𝑈 𝑋) = (𝑉 𝑋)))
61603anbi1d 1400 . . . . . . . . 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 6611 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥𝐼𝑦) = (𝑋𝐼𝑦))
6362eleq2d 2684 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑦)))
64 eleq1 2686 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑦)))
65 oveq1 6611 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥𝐼𝑧) = (𝑋𝐼𝑧))
6665eleq2d 2684 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑋𝐼𝑧)))
6763, 64, 663orbi123d 1395 . . . . . . . . . 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 746 . . . . . . . 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 6612 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑈 𝑦) = (𝑈 𝑌))
72 oveq2 6612 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑉 𝑦) = (𝑉 𝑌))
7371, 72eqeq12d 2636 . . . . . . . . . 10 (𝑦 = 𝑌 → ((𝑈 𝑦) = (𝑉 𝑦) ↔ (𝑈 𝑌) = (𝑉 𝑌)))
74733anbi2d 1401 . . . . . . . . 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 6612 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑋𝐼𝑦) = (𝑋𝐼𝑌))
7675eleq2d 2684 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑧 ∈ (𝑋𝐼𝑦) ↔ 𝑧 ∈ (𝑋𝐼𝑌)))
77 oveq2 6612 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑧𝐼𝑦) = (𝑧𝐼𝑌))
7877eleq2d 2684 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑋 ∈ (𝑧𝐼𝑦) ↔ 𝑋 ∈ (𝑧𝐼𝑌)))
79 eleq1 2686 . . . . . . . . . . 11 (𝑦 = 𝑌 → (𝑦 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑧)))
8076, 78, 793orbi123d 1395 . . . . . . . . . 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 746 . . . . . . . 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 6612 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑈 𝑧) = (𝑈 𝑍))
85 oveq2 6612 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑉 𝑧) = (𝑉 𝑍))
8684, 85eqeq12d 2636 . . . . . . . . . 10 (𝑧 = 𝑍 → ((𝑈 𝑧) = (𝑉 𝑧) ↔ (𝑈 𝑍) = (𝑉 𝑍)))
87863anbi3d 1402 . . . . . . . . 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 2686 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑧 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
89 oveq1 6611 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (𝑧𝐼𝑌) = (𝑍𝐼𝑌))
9089eleq2d 2684 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑋 ∈ (𝑧𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
91 oveq2 6612 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
9291eleq2d 2684 . . . . . . . . . . 11 (𝑧 = 𝑍 → (𝑌 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
9388, 90, 923orbi123d 1395 . . . . . . . . . 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 746 . . . . . . . 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 3309 . . . . . 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 1323 . . . . 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 438 . . . 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 384  w3o 1035  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908   class class class wbr 4613  cfv 5847  (class class class)co 6604  3c3 11015  Basecbs 15781  distcds 15871  DimTarskiGcstrkgld 25233  Itvcitv 25235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-trkgld 25251
This theorem is referenced by: (None)
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