![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 1sdom2 | Structured version Visualization version GIF version |
Description: Ordinal 1 is strictly dominated by ordinal 2. (Contributed by NM, 4-Apr-2007.) |
Ref | Expression |
---|---|
1sdom2 | ⊢ 1o ≺ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 8005 | . . 3 ⊢ 1o ∈ ω | |
2 | php4 8437 | . . 3 ⊢ (1o ∈ ω → 1o ≺ suc 1o) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 1o ≺ suc 1o |
4 | df-2o 7846 | . 2 ⊢ 2o = suc 1o | |
5 | 3, 4 | breqtrri 4915 | 1 ⊢ 1o ≺ 2o |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 class class class wbr 4888 suc csuc 5980 ωcom 7345 1oc1o 7838 2oc2o 7839 ≺ csdm 8242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-om 7346 df-1o 7845 df-2o 7846 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 |
This theorem is referenced by: pm54.43 9161 pr2ne 9163 prdom2 9164 canthp1lem1 9811 canthp1 9813 1nprm 15801 |
Copyright terms: Public domain | W3C validator |