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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 12255 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 4968 | . . 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 12681 | . . 3 ⊢ ((𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) → {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ⊆ 𝑥) |
4 | 1, 3 | mpgbir 1794 | . 2 ⊢ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ⊆ ∩ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
5 | 2 | zrei 12594 | . . . . . 6 ⊢ 𝑁 ∈ ℝ |
6 | 5 | leidi 11778 | . . . . 5 ⊢ 𝑁 ≤ 𝑁 |
7 | breq2 5152 | . . . . . 6 ⊢ (𝑧 = 𝑁 → (𝑁 ≤ 𝑧 ↔ 𝑁 ≤ 𝑁)) | |
8 | 7 | elrab 3682 | . . . . 5 ⊢ (𝑁 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ↔ (𝑁 ∈ ℤ ∧ 𝑁 ≤ 𝑁)) |
9 | 2, 6, 8 | mpbir2an 710 | . . . 4 ⊢ 𝑁 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} |
10 | peano2uz2 12680 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}) → (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}) | |
11 | 2, 10 | mpan 689 | . . . . 5 ⊢ (𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} → (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧}) |
12 | 11 | rgen 3060 | . . . 4 ⊢ ∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} |
13 | zex 12597 | . . . . . 6 ⊢ ℤ ∈ V | |
14 | 13 | rabex 5334 | . . . . 5 ⊢ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ∈ V |
15 | eleq2 2818 | . . . . . 6 ⊢ (𝑥 = {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} → (𝑁 ∈ 𝑥 ↔ 𝑁 ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧})) | |
16 | eleq2 2818 | . . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧})) | |
17 | 16 | raleqbi1dv 3330 | . . . . . 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 631 | . . . . 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 3667 | . . . 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 710 | . . 3 ⊢ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} ∈ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
21 | intss1 4966 | . . 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 3996 | 1 ⊢ {𝑧 ∈ ℤ ∣ 𝑁 ≤ 𝑧} = ∩ {𝑥 ∣ (𝑁 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {cab 2705 ∀wral 3058 {crab 3429 ⊆ wss 3947 ∩ cint 4949 class class class wbr 5148 (class class class)co 7420 1c1 11139 + caddc 11141 ≤ cle 11279 ℤcz 12588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 |
This theorem is referenced by: (None) |
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