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Mirrors > Home > MPE Home > Th. List > dfuzi | Structured version Visualization version GIF version |
Description: An expression for the upper integers that start at 𝑁 that is analogous to dfnn2 11456 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.) |
Ref | Expression |
---|---|
dfuzi.1 | ⊢ 𝑁 ∈ ℤ |
Ref | Expression |
---|---|
dfuzi | ⊢ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} = ∩ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssintab 4767 | . . 3 ⊢ ({𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ⊆ ∩ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) → {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ⊆ 𝑥)) | |
2 | dfuzi.1 | . . . 4 ⊢ 𝑁 ∈ ℤ | |
3 | 2 | peano5uzi 11887 | . . 3 ⊢ ((𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) → {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ⊆ 𝑥) |
4 | 1, 3 | mpgbir 1762 | . 2 ⊢ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ⊆ ∩ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
5 | 2 | zrei 11802 | . . . . . 6 ⊢ 𝑁 ∈ ℝ |
6 | 5 | leidi 10977 | . . . . 5 ⊢ 𝑁 ≤ 𝑁 |
7 | breq2 4934 | . . . . . 6 ⊢ (𝑧 = 𝑁 → (𝑁 ≤ 𝑧 ↔ 𝑁 ≤ 𝑁)) | |
8 | 7 | elrab 3595 | . . . . 5 ⊢ (𝑁 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ↔ (𝑁 ∈ ℤ ∧ 𝑁 ≤ 𝑁)) |
9 | 2, 6, 8 | mpbir2an 698 | . . . 4 ⊢ 𝑁 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} |
10 | peano2uz2 11886 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}) → (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}) | |
11 | 2, 10 | mpan 677 | . . . . 5 ⊢ (𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} → (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}) |
12 | 11 | rgen 3098 | . . . 4 ⊢ ∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} |
13 | zex 11805 | . . . . . 6 ⊢ ℤ ∈ V | |
14 | 13 | rabex 5092 | . . . . 5 ⊢ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ∈ V |
15 | eleq2 2854 | . . . . . 6 ⊢ (𝑥 = {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} → (𝑁 ∈ 𝑥 ↔ 𝑁 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧})) | |
16 | eleq2 2854 | . . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧})) | |
17 | 16 | raleqbi1dv 3343 | . . . . . 6 ⊢ (𝑥 = {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧})) |
18 | 15, 17 | anbi12d 621 | . . . . 5 ⊢ (𝑥 = {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} → ((𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (𝑁 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ∧ ∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}))) |
19 | 14, 18 | elab 3582 | . . . 4 ⊢ ({𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ∈ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (𝑁 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ∧ ∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧})) |
20 | 9, 12, 19 | mpbir2an 698 | . . 3 ⊢ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ∈ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
21 | intss1 4765 | . . 3 ⊢ ({𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ∈ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} → ∩ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ⊆ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}) | |
22 | 20, 21 | ax-mp 5 | . 2 ⊢ ∩ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ⊆ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} |
23 | 4, 22 | eqssi 3876 | 1 ⊢ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} = ∩ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 {cab 2758 ∀wral 3088 {crab 3092 ⊆ wss 3831 ∩ cint 4750 class class class wbr 4930 (class class class)co 6978 1c1 10338 + caddc 10340 ≤ cle 10477 ℤcz 11796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-n0 11711 df-z 11797 |
This theorem is referenced by: (None) |
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