Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > eluz | GIF version |
Description: Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.) |
Ref | Expression |
---|---|
eluz | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz1 9470 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) | |
2 | 1 | baibd 913 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2136 class class class wbr 3982 ‘cfv 5188 ≤ cle 7934 ℤcz 9191 ℤ≥cuz 9466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-cnex 7844 ax-resscn 7845 |
This theorem depends on definitions: df-bi 116 df-3or 969 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-sbc 2952 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-ov 5845 df-neg 8072 df-z 9192 df-uz 9467 |
This theorem is referenced by: uzneg 9484 uztric 9487 uzm1 9496 eluzdc 9548 fzn 9977 fzsplit2 9985 fznn 10024 uzsplit 10027 elfz2nn0 10047 fzouzsplit 10114 exfzdc 10175 fzfig 10365 faclbnd 10654 seq3coll 10755 cvg1nlemcau 10926 cvg1nlemres 10927 summodclem2a 11322 fsum0diaglem 11381 mertenslemi1 11476 prodmodclem2a 11517 zsupcllemstep 11878 zsupcl 11880 infssuzex 11882 pcpremul 12225 pcdvdsb 12251 pcadd 12271 pcfac 12280 pcbc 12281 prmunb 12292 uzdcinzz 13679 |
Copyright terms: Public domain | W3C validator |