![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > eluz2 | GIF version |
Description: Membership in an upper set of integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show 𝑀 ∈ ℤ. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
eluz2 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 8918 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
2 | simp1 939 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℤ) | |
3 | eluz1 8917 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) | |
4 | ibar 295 | . . . 4 ⊢ (𝑀 ∈ ℤ → ((𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) ↔ (𝑀 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)))) | |
5 | 3, 4 | bitrd 186 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)))) |
6 | 3anass 924 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) ↔ (𝑀 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) | |
7 | 5, 6 | syl6bbr 196 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) |
8 | 1, 2, 7 | pm5.21nii 653 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∧ w3a 920 ∈ wcel 1434 class class class wbr 3811 ‘cfv 4968 ≤ cle 7425 ℤcz 8645 ℤ≥cuz 8913 |
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-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 ax-cnex 7338 ax-resscn 7339 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-sbc 2827 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-mpt 3867 df-id 4083 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-rn 4411 df-res 4412 df-ima 4413 df-iota 4933 df-fun 4970 df-fn 4971 df-f 4972 df-fv 4976 df-ov 5593 df-neg 7558 df-z 8646 df-uz 8914 |
This theorem is referenced by: eluzuzle 8921 eluzelz 8922 eluzle 8925 uztrn 8929 eluzp1p1 8938 uznn0sub 8944 uz3m2nn 8955 1eluzge0 8956 2eluzge1 8958 raluz2 8961 rexuz2 8963 peano2uz 8965 nn0pzuz 8969 uzind4 8970 nn0ge2m1nnALT 8997 elfzuzb 9328 uzsubsubfz 9355 ige2m1fz 9416 4fvwrd4 9440 elfzo2 9450 elfzouz2 9460 fzossrbm1 9472 fzossfzop1 9511 ssfzo12bi 9524 elfzonelfzo 9529 elfzomelpfzo 9530 fzosplitprm1 9533 fzostep1 9536 fzind2 9538 flqword2 9584 fldiv4p1lem1div2 9600 ibcval5 10005 resqrexlemoverl 10280 resqrexlemga 10282 oddge22np1 10660 nn0o 10686 dvdsnprmd 10886 prmgt1 10892 oddprmgt2 10894 oddprmge3 10895 |
Copyright terms: Public domain | W3C validator |