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Mirrors > Home > MPE Home > Th. List > ordtr1 | Structured version Visualization version GIF version |
Description: Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) |
Ref | Expression |
---|---|
ordtr1 | ⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtr 6198 | . 2 ⊢ (Ord 𝐶 → Tr 𝐶) | |
2 | trel 5170 | . 2 ⊢ (Tr 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 Tr wtr 5163 Ord word 6183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-in 3940 df-ss 3949 df-uni 4831 df-tr 5164 df-ord 6187 |
This theorem is referenced by: ontr1 6230 dfsmo2 7973 smores2 7980 smoel 7986 smogt 7993 ordiso2 8967 r1ordg 9195 r1pwss 9201 r1val1 9203 rankr1ai 9215 rankval3b 9243 rankonidlem 9245 onssr1 9248 cofsmo 9679 fpwwe2lem9 10048 bnj1098 31954 bnj594 32083 nosepssdm 33087 |
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