![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > 1nen2 | GIF version |
Description: One and two are not equinumerous. (Contributed by Jim Kingdon, 25-Jan-2022.) |
Ref | Expression |
---|---|
1nen2 | ⊢ ¬ 1𝑜 ≈ 2𝑜 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 6209 | . . 3 ⊢ 1𝑜 ∈ ω | |
2 | php5 6504 | . . 3 ⊢ (1𝑜 ∈ ω → ¬ 1𝑜 ≈ suc 1𝑜) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ ¬ 1𝑜 ≈ suc 1𝑜 |
4 | df-2o 6114 | . . 3 ⊢ 2𝑜 = suc 1𝑜 | |
5 | 4 | breq2i 3819 | . 2 ⊢ (1𝑜 ≈ 2𝑜 ↔ 1𝑜 ≈ suc 1𝑜) |
6 | 3, 5 | mtbir 629 | 1 ⊢ ¬ 1𝑜 ≈ 2𝑜 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 1434 class class class wbr 3811 suc csuc 4156 ωcom 4368 1𝑜c1o 6106 2𝑜c2o 6107 ≈ cen 6385 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-tr 3902 df-id 4084 df-iord 4157 df-on 4159 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-1o 6113 df-2o 6114 df-er 6222 df-en 6388 |
This theorem is referenced by: pm54.43 6721 pr2ne 6723 1nprm 10876 |
Copyright terms: Public domain | W3C validator |