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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 6394 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
| 2 | eloni 6394 | . 2 ⊢ (𝐶 ∈ On → Ord 𝐶) | |
| 3 | ordtr2 6428 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐶) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | |
| 4 | 1, 2, 3 | syl2an 596 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 Ord word 6383 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: onelssex 6432 onunel 6489 oeordsuc 8632 oelimcl 8638 oeeui 8640 omopthlem2 8698 coflton 8709 cofon1 8710 cofon2 8711 naddssim 8723 omxpenlem 9113 oismo 9580 cantnflem1c 9727 cantnflem1 9729 cantnflem3 9731 rankr1ai 9838 rankxplim 9919 infxpenlem 10053 alephle 10128 pwcfsdom 10623 r1limwun 10776 oldbdayim 27927 addsbdaylem 28049 negsbdaylem 28088 ontopbas 36429 ontgval 36432 onexlimgt 43255 nnoeomeqom 43325 omabs2 43345 oaun3lem2 43388 nadd2rabex 43399 nadd1suc 43405 |
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