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Mirrors > Home > MPE Home > Th. List > ordtr3 | Structured version Visualization version GIF version |
Description: Transitive law for ordinal classes. (Contributed by Mario Carneiro, 30-Dec-2014.) (Proof shortened by JJ, 24-Sep-2021.) |
Ref | Expression |
---|---|
ordtr3 | ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nelss 4061 | . . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 ⊆ 𝐶) | |
2 | 1 | adantl 481 | . . . . 5 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) → ¬ 𝐵 ⊆ 𝐶) |
3 | ordtri1 6419 | . . . . . . 7 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ⊆ 𝐶 ↔ ¬ 𝐶 ∈ 𝐵)) | |
4 | 3 | con2bid 354 | . . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐶 ∈ 𝐵 ↔ ¬ 𝐵 ⊆ 𝐶)) |
5 | 4 | adantr 480 | . . . . 5 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) → (𝐶 ∈ 𝐵 ↔ ¬ 𝐵 ⊆ 𝐶)) |
6 | 2, 5 | mpbird 257 | . . . 4 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐶 ∈ 𝐵) |
7 | 6 | expr 456 | . . 3 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (¬ 𝐴 ∈ 𝐶 → 𝐶 ∈ 𝐵)) |
8 | 7 | orrd 863 | . 2 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)) |
9 | 8 | ex 412 | 1 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∈ wcel 2106 ⊆ wss 3963 Ord word 6385 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 |
This theorem is referenced by: (None) |
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