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Theorem dfuzi 9393
Description: An expression for the upper integers that start at 𝑁 that is analogous to dfnn2 8951 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
Hypothesis
Ref Expression
dfuz.1 𝑁 ∈ ℤ
Assertion
Ref Expression
dfuzi {𝑧 ∈ ℤ ∣ 𝑁𝑧} = {𝑥 ∣ (𝑁𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Distinct variable group:   𝑥,𝑦,𝑧,𝑁

Proof of Theorem dfuzi
StepHypRef Expression
1 ssintab 3876 . . 3 ({𝑧 ∈ ℤ ∣ 𝑁𝑧} ⊆ {𝑥 ∣ (𝑁𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑥((𝑁𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → {𝑧 ∈ ℤ ∣ 𝑁𝑧} ⊆ 𝑥))
2 dfuz.1 . . . 4 𝑁 ∈ ℤ
32peano5uzi 9392 . . 3 ((𝑁𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) → {𝑧 ∈ ℤ ∣ 𝑁𝑧} ⊆ 𝑥)
41, 3mpgbir 1464 . 2 {𝑧 ∈ ℤ ∣ 𝑁𝑧} ⊆ {𝑥 ∣ (𝑁𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
52zrei 9289 . . . . . 6 𝑁 ∈ ℝ
65leidi 8472 . . . . 5 𝑁𝑁
7 breq2 4022 . . . . . 6 (𝑧 = 𝑁 → (𝑁𝑧𝑁𝑁))
87elrab 2908 . . . . 5 (𝑁 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧} ↔ (𝑁 ∈ ℤ ∧ 𝑁𝑁))
92, 6, 8mpbir2an 944 . . . 4 𝑁 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}
10 peano2uz2 9390 . . . . . 6 ((𝑁 ∈ ℤ ∧ 𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}) → (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧})
112, 10mpan 424 . . . . 5 (𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧} → (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧})
1211rgen 2543 . . . 4 𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧} (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}
13 zex 9292 . . . . . 6 ℤ ∈ V
1413rabex 4162 . . . . 5 {𝑧 ∈ ℤ ∣ 𝑁𝑧} ∈ V
15 eleq2 2253 . . . . . 6 (𝑥 = {𝑧 ∈ ℤ ∣ 𝑁𝑧} → (𝑁𝑥𝑁 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
16 eleq2 2253 . . . . . . 7 (𝑥 = {𝑧 ∈ ℤ ∣ 𝑁𝑧} → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
1716raleqbi1dv 2694 . . . . . 6 (𝑥 = {𝑧 ∈ ℤ ∣ 𝑁𝑧} → (∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧} (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
1815, 17anbi12d 473 . . . . 5 (𝑥 = {𝑧 ∈ ℤ ∣ 𝑁𝑧} → ((𝑁𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (𝑁 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧} ∧ ∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧} (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧})))
1914, 18elab 2896 . . . 4 ({𝑧 ∈ ℤ ∣ 𝑁𝑧} ∈ {𝑥 ∣ (𝑁𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ (𝑁 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧} ∧ ∀𝑦 ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧} (𝑦 + 1) ∈ {𝑧 ∈ ℤ ∣ 𝑁𝑧}))
209, 12, 19mpbir2an 944 . . 3 {𝑧 ∈ ℤ ∣ 𝑁𝑧} ∈ {𝑥 ∣ (𝑁𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
21 intss1 3874 . . 3 ({𝑧 ∈ ℤ ∣ 𝑁𝑧} ∈ {𝑥 ∣ (𝑁𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} → {𝑥 ∣ (𝑁𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ⊆ {𝑧 ∈ ℤ ∣ 𝑁𝑧})
2220, 21ax-mp 5 . 2 {𝑥 ∣ (𝑁𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)} ⊆ {𝑧 ∈ ℤ ∣ 𝑁𝑧}
234, 22eqssi 3186 1 {𝑧 ∈ ℤ ∣ 𝑁𝑧} = {𝑥 ∣ (𝑁𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  {cab 2175  wral 2468  {crab 2472  wss 3144   cint 3859   class class class wbr 4018  (class class class)co 5896  1c1 7842   + caddc 7844  cle 8023  cz 9283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-ltadd 7957
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-inn 8950  df-n0 9207  df-z 9284
This theorem is referenced by: (None)
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