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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 6319 | . . 3 ⊢ 1o ∈ ω | |
2 | php5 6654 | . . 3 ⊢ (1o ∈ ω → ¬ 1o ≈ suc 1o) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ ¬ 1o ≈ suc 1o |
4 | df-2o 6220 | . . 3 ⊢ 2o = suc 1o | |
5 | 4 | breq2i 3875 | . 2 ⊢ (1o ≈ 2o ↔ 1o ≈ suc 1o) |
6 | 3, 5 | mtbir 634 | 1 ⊢ ¬ 1o ≈ 2o |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 1445 class class class wbr 3867 suc csuc 4216 ωcom 4433 1oc1o 6212 2oc2o 6213 ≈ cen 6535 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-tr 3959 df-id 4144 df-iord 4217 df-on 4219 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-1o 6219 df-2o 6220 df-er 6332 df-en 6538 |
This theorem is referenced by: pm54.43 6915 pr2ne 6917 1nprm 11538 |
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