Theorem nzrnz 20252

Description: One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)

Hypotheses

Ref	Expression
isnzr.o	⊢ 1 = (1r‘𝑅)
isnzr.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
nzrnz	⊢ (𝑅 ∈ NzRing → 1 ≠ 0 )

Proof of Theorem nzrnz

Step	Hyp	Ref	Expression
1		isnzr.o	. . 3 ⊢ 1 = (1r‘𝑅)
2		isnzr.z	. . 3 ⊢ 0 = (0g‘𝑅)
3	1, 2	isnzr 20251	. 2 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ))
4	3	simprbi 500	1 ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 )

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2112   ≠ wne 2932  ‘cfv 6358  0_gc0g 16898  1_rcur 19470  Ringcrg 19516  NzRingcnzr 20249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ne 2933  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-iota 6316  df-fv 6366  df-nzr 20250

This theorem is referenced by:  nzrunit  20259  subrgnzr  20260  fidomndrng  20299  uvcf1  20708  lindfind2  20734  nm1  23519  deg1pw  24972  ply1nz  24973  ply1nzb  24974  lgsqrlem4  26184  zrhnm  31585  uvcn0  39918  mon1pid  40674  deg1mhm  40676  nrhmzr  45047