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Mirrors > Home > ILE Home > Th. List > zapne | GIF version |
Description: Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Ref | Expression |
---|---|
zapne | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁 ↔ 𝑀 ≠ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 8356 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
2 | zcn 8356 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
3 | apne 7723 | . . 3 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑀 # 𝑁 → 𝑀 ≠ 𝑁)) | |
4 | 1, 2, 3 | syl2an 283 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁 → 𝑀 ≠ 𝑁)) |
5 | df-ne 2246 | . . 3 ⊢ (𝑀 ≠ 𝑁 ↔ ¬ 𝑀 = 𝑁) | |
6 | ztri3or 8394 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀)) | |
7 | 3orrot 925 | . . . . . . 7 ⊢ ((𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀) ↔ (𝑀 = 𝑁 ∨ 𝑁 < 𝑀 ∨ 𝑀 < 𝑁)) | |
8 | 3orass 922 | . . . . . . 7 ⊢ ((𝑀 = 𝑁 ∨ 𝑁 < 𝑀 ∨ 𝑀 < 𝑁) ↔ (𝑀 = 𝑁 ∨ (𝑁 < 𝑀 ∨ 𝑀 < 𝑁))) | |
9 | 7, 8 | bitri 182 | . . . . . 6 ⊢ ((𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀) ↔ (𝑀 = 𝑁 ∨ (𝑁 < 𝑀 ∨ 𝑀 < 𝑁))) |
10 | 6, 9 | sylib 120 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 = 𝑁 ∨ (𝑁 < 𝑀 ∨ 𝑀 < 𝑁))) |
11 | 10 | ord 675 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑀 = 𝑁 → (𝑁 < 𝑀 ∨ 𝑀 < 𝑁))) |
12 | zre 8355 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
13 | zre 8355 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
14 | reaplt 7688 | . . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 # 𝑁 ↔ (𝑀 < 𝑁 ∨ 𝑁 < 𝑀))) | |
15 | orcom 679 | . . . . . 6 ⊢ ((𝑀 < 𝑁 ∨ 𝑁 < 𝑀) ↔ (𝑁 < 𝑀 ∨ 𝑀 < 𝑁)) | |
16 | 14, 15 | syl6bb 194 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 # 𝑁 ↔ (𝑁 < 𝑀 ∨ 𝑀 < 𝑁))) |
17 | 12, 13, 16 | syl2an 283 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁 ↔ (𝑁 < 𝑀 ∨ 𝑀 < 𝑁))) |
18 | 11, 17 | sylibrd 167 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑀 = 𝑁 → 𝑀 # 𝑁)) |
19 | 5, 18 | syl5bi 150 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≠ 𝑁 → 𝑀 # 𝑁)) |
20 | 4, 19 | impbid 127 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁 ↔ 𝑀 ≠ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 661 ∨ w3o 918 = wceq 1284 ∈ wcel 1433 ≠ wne 2245 class class class wbr 3785 ℂcc 6979 ℝcr 6980 < clt 7153 # cap 7681 ℤcz 8351 |
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-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 |
This theorem depends on definitions: df-bi 115 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-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 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-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-inn 8040 df-n0 8289 df-z 8352 |
This theorem is referenced by: zltlen 8426 msqznn 8447 qapne 8724 qreccl 8727 nn0opthd 9649 nnabscl 9986 dvdsval2 10198 dvdscmulr 10224 dvdsmulcr 10225 divconjdvds 10249 gcdn0gt0 10369 lcmcllem 10449 lcmid 10462 3lcm2e6woprm 10468 6lcm4e12 10469 mulgcddvds 10476 divgcdcoprmex 10484 cncongr1 10485 cncongr2 10486 isprm3 10500 |
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