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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 3978 | . . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 ⊆ 𝐶) | |
2 | 1 | adantl 485 | . . . . 5 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) → ¬ 𝐵 ⊆ 𝐶) |
3 | ordtri1 6192 | . . . . . . 7 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ⊆ 𝐶 ↔ ¬ 𝐶 ∈ 𝐵)) | |
4 | 3 | con2bid 358 | . . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐶 ∈ 𝐵 ↔ ¬ 𝐵 ⊆ 𝐶)) |
5 | 4 | adantr 484 | . . . . 5 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) → (𝐶 ∈ 𝐵 ↔ ¬ 𝐵 ⊆ 𝐶)) |
6 | 2, 5 | mpbird 260 | . . . 4 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) → 𝐶 ∈ 𝐵) |
7 | 6 | expr 460 | . . 3 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (¬ 𝐴 ∈ 𝐶 → 𝐶 ∈ 𝐵)) |
8 | 7 | orrd 860 | . 2 ⊢ (((Ord 𝐵 ∧ Ord 𝐶) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)) |
9 | 8 | ex 416 | 1 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∈ wcel 2111 ⊆ wss 3881 Ord word 6158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 |
This theorem is referenced by: (None) |
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