![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > onss | Structured version Visualization version GIF version |
Description: An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.) |
Ref | Expression |
---|---|
onss | ⊢ (𝐴 ∈ On → 𝐴 ⊆ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6385 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordsson 7790 | . 2 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ⊆ wss 3946 Ord word 6374 Oncon0 6375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pr 5432 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-tr 5270 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-ord 6378 df-on 6379 |
This theorem is referenced by: predonOLD 7794 onuni 7796 onminex 7810 sucexeloniOLD 7818 suceloniOLD 7820 onssi 7846 tfi 7862 soseq 8172 tfr3 8428 tz7.49 8474 tz7.49c 8475 oacomf1olem 8593 oeeulem 8630 cofonr 8703 naddcllem 8705 naddov2 8708 naddunif 8722 naddasslem1 8723 naddasslem2 8724 ordtypelem2 9558 cantnfcl 9706 cantnflt 9711 cantnfp1lem3 9719 oemapvali 9723 cantnflem1c 9726 cantnflem1d 9727 cantnflem1 9728 cantnf 9732 cnfcom 9739 cnfcom3lem 9742 infxpenlem 10052 ac10ct 10073 dfac12lem1 10182 dfac12lem2 10183 cfeq0 10295 cfsuc 10296 cff1 10297 cfflb 10298 cofsmo 10308 cfsmolem 10309 alephsing 10315 zorn2lem2 10536 ttukeylem3 10550 ttukeylem5 10552 ttukeylem6 10553 inar1 10814 nosupno 27725 elold 27885 ontgval 36091 aomclem6 42657 tfsconcatlem 42939 tfsconcatfv 42944 ofoafo 42959 ofoaid1 42961 ofoaid2 42962 dfno2 43032 |
Copyright terms: Public domain | W3C validator |