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Mirrors > Home > MPE Home > Th. List > ontr | Structured version Visualization version GIF version |
Description: An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.) Put in closed form. (Resised by BJ, 28-Dec-2024.) |
Ref | Expression |
---|---|
ontr | ⊢ (𝐴 ∈ On → Tr 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6371 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordtr 6375 | . 2 ⊢ (Ord 𝐴 → Tr 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ On → Tr 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Tr wtr 5264 Ord word 6360 Oncon0 6361 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-v 3476 df-in 3954 df-ss 3964 df-uni 4908 df-tr 5265 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-ord 6364 df-on 6365 |
This theorem is referenced by: onunisuc 6471 ontrciOLD 6473 onuninsuci 7825 hfuni 35144 |
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