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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 6199 | . 2 ⊢ (Ord 𝐶 → Tr 𝐶) | |
2 | trel 5171 | . 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 398 ∈ wcel 2110 Tr wtr 5164 Ord word 6184 |
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-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3942 df-ss 3951 df-uni 4832 df-tr 5165 df-ord 6188 |
This theorem is referenced by: ontr1 6231 dfsmo2 7978 smores2 7985 smoel 7991 smogt 7998 ordiso2 8973 r1ordg 9201 r1pwss 9207 r1val1 9209 rankr1ai 9221 rankval3b 9249 rankonidlem 9251 onssr1 9254 cofsmo 9685 fpwwe2lem9 10054 bnj1098 32050 bnj594 32179 nosepssdm 33185 |
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