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Mirrors > Home > ILE Home > Th. List > eluz2b1 | GIF version |
Description: Two ways to say "an integer greater than or equal to 2". (Contributed by Paul Chapman, 23-Nov-2012.) |
Ref | Expression |
---|---|
eluz2b1 | ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 9312 | . . 3 ⊢ 2 ∈ ℤ | |
2 | 1 | eluz1i 9566 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 2 ≤ 𝑁)) |
3 | 1z 9310 | . . . . 5 ⊢ 1 ∈ ℤ | |
4 | zltp1le 9338 | . . . . 5 ⊢ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1 < 𝑁 ↔ (1 + 1) ≤ 𝑁)) | |
5 | 3, 4 | mpan 424 | . . . 4 ⊢ (𝑁 ∈ ℤ → (1 < 𝑁 ↔ (1 + 1) ≤ 𝑁)) |
6 | df-2 9009 | . . . . 5 ⊢ 2 = (1 + 1) | |
7 | 6 | breq1i 4025 | . . . 4 ⊢ (2 ≤ 𝑁 ↔ (1 + 1) ≤ 𝑁) |
8 | 5, 7 | bitr4di 198 | . . 3 ⊢ (𝑁 ∈ ℤ → (1 < 𝑁 ↔ 2 ≤ 𝑁)) |
9 | 8 | pm5.32i 454 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 1 < 𝑁) ↔ (𝑁 ∈ ℤ ∧ 2 ≤ 𝑁)) |
10 | 2, 9 | bitr4i 187 | 1 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∈ wcel 2160 class class class wbr 4018 ‘cfv 5235 (class class class)co 5897 1c1 7843 + caddc 7845 < clt 8023 ≤ cle 8024 2c2 9001 ℤcz 9284 ℤ≥cuz 9559 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-ltadd 7958 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-inn 8951 df-2 9009 df-n0 9208 df-z 9285 df-uz 9560 |
This theorem is referenced by: eluz2gt1 9634 eluz2b2 9635 uz2m1nn 9637 uz2mulcl 9640 prmind2 12155 2prm 12162 3prm 12163 sqnprm 12171 isprm5lem 12176 difsqpwdvds 12373 exmidunben 12480 |
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