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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 9484 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
| 2 | zcn 9484 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
| 3 | apne 8803 | . . 3 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑀 # 𝑁 → 𝑀 ≠ 𝑁)) | |
| 4 | 1, 2, 3 | syl2an 289 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁 → 𝑀 ≠ 𝑁)) |
| 5 | df-ne 2403 | . . 3 ⊢ (𝑀 ≠ 𝑁 ↔ ¬ 𝑀 = 𝑁) | |
| 6 | ztri3or 9522 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀)) | |
| 7 | 3orrot 1010 | . . . . . . 7 ⊢ ((𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀) ↔ (𝑀 = 𝑁 ∨ 𝑁 < 𝑀 ∨ 𝑀 < 𝑁)) | |
| 8 | 3orass 1007 | . . . . . . 7 ⊢ ((𝑀 = 𝑁 ∨ 𝑁 < 𝑀 ∨ 𝑀 < 𝑁) ↔ (𝑀 = 𝑁 ∨ (𝑁 < 𝑀 ∨ 𝑀 < 𝑁))) | |
| 9 | 7, 8 | bitri 184 | . . . . . 6 ⊢ ((𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀) ↔ (𝑀 = 𝑁 ∨ (𝑁 < 𝑀 ∨ 𝑀 < 𝑁))) |
| 10 | 6, 9 | sylib 122 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 = 𝑁 ∨ (𝑁 < 𝑀 ∨ 𝑀 < 𝑁))) |
| 11 | 10 | ord 731 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑀 = 𝑁 → (𝑁 < 𝑀 ∨ 𝑀 < 𝑁))) |
| 12 | zre 9483 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
| 13 | zre 9483 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
| 14 | reaplt 8768 | . . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 # 𝑁 ↔ (𝑀 < 𝑁 ∨ 𝑁 < 𝑀))) | |
| 15 | orcom 735 | . . . . . 6 ⊢ ((𝑀 < 𝑁 ∨ 𝑁 < 𝑀) ↔ (𝑁 < 𝑀 ∨ 𝑀 < 𝑁)) | |
| 16 | 14, 15 | bitrdi 196 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 # 𝑁 ↔ (𝑁 < 𝑀 ∨ 𝑀 < 𝑁))) |
| 17 | 12, 13, 16 | syl2an 289 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁 ↔ (𝑁 < 𝑀 ∨ 𝑀 < 𝑁))) |
| 18 | 11, 17 | sylibrd 169 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑀 = 𝑁 → 𝑀 # 𝑁)) |
| 19 | 5, 18 | biimtrid 152 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≠ 𝑁 → 𝑀 # 𝑁)) |
| 20 | 4, 19 | impbid 129 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 # 𝑁 ↔ 𝑀 ≠ 𝑁)) |
| Colors of variables: wff set class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 715 ∨ w3o 1003 = wceq 1397 ∈ wcel 2202 ≠ wne 2402 class class class wbr 4088 ℂcc 8030 ℝcr 8031 < clt 8214 # cap 8761 ℤcz 9479 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 |
| This theorem depends on definitions: df-bi 117 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-br 4089 df-opab 4151 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-iota 5286 df-fun 5328 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-inn 9144 df-n0 9403 df-z 9480 |
| This theorem is referenced by: zltlen 9558 msqznn 9580 qapne 9873 qreccl 9876 seqf1oglem1 10782 nn0opthd 10985 fihashneq0 11057 nnabscl 11662 eftcl 12217 dvdsval2 12353 dvdscmulr 12383 dvdsmulcr 12384 fsumdvds 12405 divconjdvds 12412 gcdn0gt0 12551 lcmcllem 12641 lcmid 12654 3lcm2e6woprm 12660 6lcm4e12 12661 mulgcddvds 12668 divgcdcoprmex 12676 cncongr1 12677 cncongr2 12678 isprm3 12692 pcpremul 12868 pceu 12870 pcmul 12876 pcdiv 12877 pcqmul 12878 dvdsprmpweqle 12912 qexpz 12927 4sqlem11 12976 relogbval 15678 relogbzcl 15679 nnlogbexp 15686 logbgcd1irraplemexp 15695 lgslem1 15732 lgsdilem2 15768 lgsdi 15769 lgsne0 15770 lgseisen 15806 |
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