![]() |
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 7873 | . . 3 ⊢ 1𝑜 ∈ ω | |
2 | php4 8303 | . . 3 ⊢ (1𝑜 ∈ ω → 1𝑜 ≺ suc 1𝑜) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 1𝑜 ≺ suc 1𝑜 |
4 | df-2o 7714 | . 2 ⊢ 2𝑜 = suc 1𝑜 | |
5 | 3, 4 | breqtrri 4813 | 1 ⊢ 1𝑜 ≺ 2𝑜 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2145 class class class wbr 4786 suc csuc 5868 ωcom 7212 1𝑜c1o 7706 2𝑜c2o 7707 ≺ csdm 8108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-om 7213 df-1o 7713 df-2o 7714 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 |
This theorem is referenced by: pm54.43 9026 pr2ne 9028 prdom2 9029 canthp1lem1 9676 canthp1 9678 1nprm 15599 |
Copyright terms: Public domain | W3C validator |