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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 6116 | . . 3 ⊢ 1𝑜 ∈ ω | |
2 | php5 6344 | . . 3 ⊢ (1𝑜 ∈ ω → ¬ 1𝑜 ≈ suc 1𝑜) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ ¬ 1𝑜 ≈ suc 1𝑜 |
4 | df-2o 6025 | . . 3 ⊢ 2𝑜 = suc 1𝑜 | |
5 | 4 | breq2i 3793 | . 2 ⊢ (1𝑜 ≈ 2𝑜 ↔ 1𝑜 ≈ suc 1𝑜) |
6 | 3, 5 | mtbir 628 | 1 ⊢ ¬ 1𝑜 ≈ 2𝑜 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 1433 class class class wbr 3785 suc csuc 4120 ωcom 4331 1𝑜c1o 6017 2𝑜c2o 6018 ≈ cen 6242 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-1o 6024 df-2o 6025 df-er 6129 df-en 6245 |
This theorem is referenced by: pm54.43 6459 pr2ne 6461 1nprm 10496 |
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