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Mirrors > Home > MPE Home > Th. List > ordtr3OLD | Structured version Visualization version GIF version |
Description: Obsolete proof of ordtr3 5930 as of 24-Sep-2021. (Contributed by Mario Carneiro, 30-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
ordtr3OLD | ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 479 | . . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → Ord 𝐶) | |
2 | ordelord 5906 | . . . . . 6 ⊢ ((Ord 𝐵 ∧ 𝐴 ∈ 𝐵) → Ord 𝐴) | |
3 | 2 | adantlr 753 | . . . . 5 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → Ord 𝐴) |
4 | ordtri1 5917 | . . . . 5 ⊢ ((Ord 𝐶 ∧ Ord 𝐴) → (𝐶 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ 𝐶)) | |
5 | 1, 3, 4 | syl2an2r 911 | . . . 4 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (𝐶 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ 𝐶)) |
6 | ordtr2 5929 | . . . . . . 7 ⊢ ((Ord 𝐶 ∧ Ord 𝐵) → ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐶 ∈ 𝐵)) | |
7 | 6 | ancoms 468 | . . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐶 ∈ 𝐵)) |
8 | 7 | expcomd 453 | . . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ⊆ 𝐴 → 𝐶 ∈ 𝐵))) |
9 | 8 | imp 444 | . . . 4 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (𝐶 ⊆ 𝐴 → 𝐶 ∈ 𝐵)) |
10 | 5, 9 | sylbird 250 | . . 3 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (¬ 𝐴 ∈ 𝐶 → 𝐶 ∈ 𝐵)) |
11 | 10 | orrd 392 | . 2 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)) |
12 | 11 | ex 449 | 1 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 382 ∧ wa 383 ∈ wcel 2139 ⊆ wss 3715 Ord word 5883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-tr 4905 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-ord 5887 |
This theorem is referenced by: (None) |
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