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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. (Contributed by NM, 11-Aug-1994.) |
Ref | Expression |
---|---|
ontr1 | ⊢ (𝐶 ∈ On → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 5894 | . 2 ⊢ (𝐶 ∈ On → Ord 𝐶) | |
2 | ordtr1 5928 | . 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 383 ∈ wcel 2139 Ord word 5883 Oncon0 5884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-v 3342 df-in 3722 df-ss 3729 df-uni 4589 df-tr 4905 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-ord 5887 df-on 5888 |
This theorem is referenced by: smoiun 7627 dif20el 7754 oeordi 7836 omabs 7896 omsmolem 7902 cofsmo 9283 cfsmolem 9284 inar1 9789 grur1a 9833 nosupno 32155 nosupbnd2lem1 32167 |
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