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Theorem 1nn 7666
Description: Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
Assertion
Ref Expression
1nn 1

Proof of Theorem 1nn
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfnn2 7657 . . . 4 ℕ = {x ∣ (1 x y x (y + 1) x)}
21eleq2i 2101 . . 3 (1 ℕ ↔ 1 {x ∣ (1 x y x (y + 1) x)})
3 1re 6784 . . . 4 1
4 elintg 3614 . . . 4 (1 ℝ → (1 {x ∣ (1 x y x (y + 1) x)} ↔ z {x ∣ (1 x y x (y + 1) x)}1 z))
53, 4ax-mp 7 . . 3 (1 {x ∣ (1 x y x (y + 1) x)} ↔ z {x ∣ (1 x y x (y + 1) x)}1 z)
62, 5bitri 173 . 2 (1 ℕ ↔ z {x ∣ (1 x y x (y + 1) x)}1 z)
7 vex 2554 . . . 4 z V
8 eleq2 2098 . . . . 5 (x = z → (1 x ↔ 1 z))
9 eleq2 2098 . . . . . 6 (x = z → ((y + 1) x ↔ (y + 1) z))
109raleqbi1dv 2507 . . . . 5 (x = z → (y x (y + 1) xy z (y + 1) z))
118, 10anbi12d 442 . . . 4 (x = z → ((1 x y x (y + 1) x) ↔ (1 z y z (y + 1) z)))
127, 11elab 2681 . . 3 (z {x ∣ (1 x y x (y + 1) x)} ↔ (1 z y z (y + 1) z))
1312simplbi 259 . 2 (z {x ∣ (1 x y x (y + 1) x)} → 1 z)
146, 13mprgbir 2373 1 1
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   wcel 1390  {cab 2023  wral 2300   cint 3606  (class class class)co 5455  cr 6670  1c1 6672   + caddc 6674  cn 7655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-1re 6737
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-int 3607  df-inn 7656
This theorem is referenced by:  nnind  7671  nn1suc  7674  2nn  7815  1nn0  7933  nn0p1nn  7957  1z  8007  neg1z  8013  elz2  8048  nneoor  8076  indstr  8272  elnn1uz2  8280  zq  8297  qreccl  8311  expivallem  8870  exp1  8875  nnexpcl  8882  expnbnd  8985
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