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Mirrors > Home > MPE Home > Th. List > po2ne | Structured version Visualization version GIF version |
Description: Two classes which are in a partial order relation are not equal. (Contributed by AV, 13-Mar-2023.) |
Ref | Expression |
---|---|
po2ne | ⊢ ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴𝑅𝐵) → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5068 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐵)) | |
2 | poirr 5484 | . . . . . . . . 9 ⊢ ((𝑅 Po 𝑉 ∧ 𝐵 ∈ 𝑉) → ¬ 𝐵𝑅𝐵) | |
3 | 2 | adantrl 714 | . . . . . . . 8 ⊢ ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ¬ 𝐵𝑅𝐵) |
4 | 3 | pm2.21d 121 | . . . . . . 7 ⊢ ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐵𝑅𝐵 → 𝐴 ≠ 𝐵)) |
5 | 4 | ex 415 | . . . . . 6 ⊢ (𝑅 Po 𝑉 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐵𝑅𝐵 → 𝐴 ≠ 𝐵))) |
6 | 5 | com13 88 | . . . . 5 ⊢ (𝐵𝑅𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑅 Po 𝑉 → 𝐴 ≠ 𝐵))) |
7 | 1, 6 | syl6bi 255 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑅 Po 𝑉 → 𝐴 ≠ 𝐵)))) |
8 | 7 | com24 95 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑅 Po 𝑉 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴𝑅𝐵 → 𝐴 ≠ 𝐵)))) |
9 | 8 | 3impd 1344 | . 2 ⊢ (𝐴 = 𝐵 → ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴𝑅𝐵) → 𝐴 ≠ 𝐵)) |
10 | ax-1 6 | . 2 ⊢ (𝐴 ≠ 𝐵 → ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴𝑅𝐵) → 𝐴 ≠ 𝐵)) | |
11 | 9, 10 | pm2.61ine 3100 | 1 ⊢ ((𝑅 Po 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴𝑅𝐵) → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5065 Po wpo 5471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-po 5473 |
This theorem is referenced by: prproropf1olem1 43664 |
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