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Mirrors > Home > ILE Home > Th. List > 2eluzge1 | GIF version |
Description: 2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
Ref | Expression |
---|---|
2eluzge1 | ⊢ 2 ∈ (ℤ≥‘1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 8874 | . 2 ⊢ 1 ∈ ℤ | |
2 | 2z 8876 | . 2 ⊢ 2 ∈ ℤ | |
3 | 1le2 8722 | . 2 ⊢ 1 ≤ 2 | |
4 | eluz2 9124 | . 2 ⊢ (2 ∈ (ℤ≥‘1) ↔ (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 1 ≤ 2)) | |
5 | 1, 2, 3, 4 | mpbir3an 1128 | 1 ⊢ 2 ∈ (ℤ≥‘1) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1445 class class class wbr 3867 ‘cfv 5049 1c1 7448 ≤ cle 7620 2c2 8571 ℤcz 8848 ℤ≥cuz 9118 |
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-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-addass 7544 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-ltadd 7558 |
This theorem depends on definitions: df-bi 116 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-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 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-mpt 3923 df-id 4144 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-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-inn 8521 df-2 8579 df-z 8849 df-uz 9119 |
This theorem is referenced by: isprm3 11543 |
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