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Mirrors > Home > MPE Home > Th. List > otthne | Structured version Visualization version GIF version |
Description: Contrapositive of the ordered triple theorem. (Contributed by Scott Fenton, 31-Jan-2025.) |
Ref | Expression |
---|---|
otthne.1 | ⊢ 𝐴 ∈ V |
otthne.2 | ⊢ 𝐵 ∈ V |
otthne.3 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
otthne | ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | otthne.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
2 | otthne.2 | . . . . 5 ⊢ 𝐵 ∈ V | |
3 | otthne.3 | . . . . 5 ⊢ 𝐶 ∈ V | |
4 | 1, 2, 3 | otth 5485 | . . . 4 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹)) |
5 | 4 | notbii 320 | . . 3 ⊢ (¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ ¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹)) |
6 | 3ianor 1108 | . . 3 ⊢ (¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹) ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹)) | |
7 | 5, 6 | bitri 275 | . 2 ⊢ (¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹)) |
8 | df-ne 2942 | . 2 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ ¬ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩) | |
9 | df-ne 2942 | . . 3 ⊢ (𝐴 ≠ 𝐷 ↔ ¬ 𝐴 = 𝐷) | |
10 | df-ne 2942 | . . 3 ⊢ (𝐵 ≠ 𝐸 ↔ ¬ 𝐵 = 𝐸) | |
11 | df-ne 2942 | . . 3 ⊢ (𝐶 ≠ 𝐹 ↔ ¬ 𝐶 = 𝐹) | |
12 | 9, 10, 11 | 3orbi123i 1157 | . 2 ⊢ ((𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹) ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐸 ∨ ¬ 𝐶 = 𝐹)) |
13 | 7, 8, 12 | 3bitr4i 303 | 1 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∨ w3o 1087 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ⟨cotp 4637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-ot 4638 |
This theorem is referenced by: xpord3lem 8135 xpord3pred 8138 xpord3inddlem 8140 |
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