Theorem nzrring 13280

Description: A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.)

Assertion

Ref	Expression
nzrring	⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)

Proof of Theorem nzrring

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nzr 13277	. . 3 ⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}
2	1	ssrab3 3241	. 2 ⊢ NzRing ⊆ Ring
3	2	sseli 3151	1 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2148   ≠ wne 2347  ‘cfv 5216  0_gc0g 12695  1_rcur 13095  Ringcrg 13132  NzRingcnzr 13276

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-in 3135  df-ss 3142  df-nzr 13277

This theorem is referenced by:  nzrunit  13282  lringring  13288