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Theorem nn0suc 4252
 Description: A natural number is either 0 or a successor. Similar theorems for arbitrary sets or real numbers will not be provable (without the law of the excluded middle), but equality of natural numbers is decidable. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
nn0suc (A 𝜔 → (A = ∅ x 𝜔 A = suc x))
Distinct variable group:   x,A

Proof of Theorem nn0suc
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2029 . . 3 (y = ∅ → (y = ∅ ↔ ∅ = ∅))
2 eqeq1 2029 . . . 4 (y = ∅ → (y = suc x ↔ ∅ = suc x))
32rexbidv 2304 . . 3 (y = ∅ → (x 𝜔 y = suc xx 𝜔 ∅ = suc x))
41, 3orbi12d 694 . 2 (y = ∅ → ((y = ∅ x 𝜔 y = suc x) ↔ (∅ = ∅ x 𝜔 ∅ = suc x)))
5 eqeq1 2029 . . 3 (y = z → (y = ∅ ↔ z = ∅))
6 eqeq1 2029 . . . 4 (y = z → (y = suc xz = suc x))
76rexbidv 2304 . . 3 (y = z → (x 𝜔 y = suc xx 𝜔 z = suc x))
85, 7orbi12d 694 . 2 (y = z → ((y = ∅ x 𝜔 y = suc x) ↔ (z = ∅ x 𝜔 z = suc x)))
9 eqeq1 2029 . . 3 (y = suc z → (y = ∅ ↔ suc z = ∅))
10 eqeq1 2029 . . . 4 (y = suc z → (y = suc x ↔ suc z = suc x))
1110rexbidv 2304 . . 3 (y = suc z → (x 𝜔 y = suc xx 𝜔 suc z = suc x))
129, 11orbi12d 694 . 2 (y = suc z → ((y = ∅ x 𝜔 y = suc x) ↔ (suc z = ∅ x 𝜔 suc z = suc x)))
13 eqeq1 2029 . . 3 (y = A → (y = ∅ ↔ A = ∅))
14 eqeq1 2029 . . . 4 (y = A → (y = suc xA = suc x))
1514rexbidv 2304 . . 3 (y = A → (x 𝜔 y = suc xx 𝜔 A = suc x))
1613, 15orbi12d 694 . 2 (y = A → ((y = ∅ x 𝜔 y = suc x) ↔ (A = ∅ x 𝜔 A = suc x)))
17 eqid 2023 . . 3 ∅ = ∅
1817orci 637 . 2 (∅ = ∅ x 𝜔 ∅ = suc x)
19 eqid 2023 . . . . 5 suc z = suc z
20 suceq 4086 . . . . . . 7 (x = z → suc x = suc z)
2120eqeq2d 2034 . . . . . 6 (x = z → (suc z = suc x ↔ suc z = suc z))
2221rspcev 2632 . . . . 5 ((z 𝜔 suc z = suc z) → x 𝜔 suc z = suc x)
2319, 22mpan2 403 . . . 4 (z 𝜔 → x 𝜔 suc z = suc x)
2423olcd 640 . . 3 (z 𝜔 → (suc z = ∅ x 𝜔 suc z = suc x))
2524a1d 22 . 2 (z 𝜔 → ((z = ∅ x 𝜔 z = suc x) → (suc z = ∅ x 𝜔 suc z = suc x)))
264, 8, 12, 16, 18, 25finds 4248 1 (A 𝜔 → (A = ∅ x 𝜔 A = suc x))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 616   = wceq 1228   ∈ wcel 1375  ∃wrex 2284  ∅c0 3200  suc csuc 4049  𝜔com 4238 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-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-nul 3856  ax-pow 3900  ax-pr 3917  ax-un 4118  ax-iinf 4236 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-dif 2896  df-un 2898  df-in 2900  df-ss 2907  df-nul 3201  df-pw 3335  df-sn 3355  df-pr 3356  df-uni 3554  df-int 3589  df-suc 4055  df-iom 4239 This theorem is referenced by:  nnsuc  4263
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