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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 6276 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | eloni 6276 | . 2 ⊢ (𝐶 ∈ On → Ord 𝐶) | |
3 | ordtr2 6310 | . 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 396 ∈ wcel 2106 ⊆ wss 3887 Ord word 6265 Oncon0 6266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-11 2154 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-tr 5192 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-ord 6269 df-on 6270 |
This theorem is referenced by: oeordsuc 8425 oelimcl 8431 oeeui 8433 omopthlem2 8490 omxpenlem 8860 oismo 9299 cantnflem1c 9445 cantnflem1 9447 cantnflem3 9449 rankr1ai 9556 rankxplim 9637 infxpenlem 9769 alephle 9844 pwcfsdom 10339 r1limwun 10492 onelssex 33661 onunel 33689 naddssim 33837 oldbdayim 34071 ontopbas 34617 ontgval 34620 |
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