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Mirrors > Home > MPE Home > Th. List > ontr2 | Structured version Visualization version GIF version |
Description: Transitive law for ordinal numbers. Exercise 3 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Nov-2003.) |
Ref | Expression |
---|---|
ontr2 | ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6169 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | eloni 6169 | . 2 ⊢ (𝐶 ∈ On → Ord 𝐶) | |
3 | ordtr2 6203 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐶) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | |
4 | 1, 2, 3 | syl2an 598 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ⊆ wss 3881 Ord word 6158 Oncon0 6159 |
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 df-on 6163 |
This theorem is referenced by: oeordsuc 8203 oelimcl 8209 oeeui 8211 omopthlem2 8266 omxpenlem 8601 oismo 8988 cantnflem1c 9134 cantnflem1 9136 cantnflem3 9138 rankr1ai 9211 rankxplim 9292 infxpenlem 9424 alephle 9499 pwcfsdom 9994 r1limwun 10147 ontopbas 33889 ontgval 33892 |
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