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Mirrors > Home > MPE Home > Th. List > ontr1 | Structured version Visualization version GIF version |
Description: Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. Theorem 1.9(ii) of [Schloeder] p. 1. (Contributed by NM, 11-Aug-1994.) |
Ref | Expression |
---|---|
ontr1 | ⊢ (𝐶 ∈ On → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6396 | . 2 ⊢ (𝐶 ∈ On → Ord 𝐶) | |
2 | ordtr1 6429 | . 2 ⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐶 ∈ On → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 Ord word 6385 Oncon0 6386 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-v 3480 df-ss 3980 df-uni 4913 df-tr 5266 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 |
This theorem is referenced by: epweon 7794 smoiun 8400 dif20el 8542 oeordi 8624 omabs 8688 omsmolem 8694 naddel12 8737 naddsuc2 8738 cofsmo 10307 cfsmolem 10308 inar1 10813 grur1a 10857 nosupno 27763 nosupbnd2lem1 27775 noinfno 27778 noinfbnd2lem1 27790 lrrecpo 27989 addsproplem2 28018 onexoegt 43233 oneltr 43245 oaun3lem1 43364 nadd2rabtr 43374 naddwordnexlem0 43386 oawordex3 43390 naddwordnexlem4 43391 |
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