Theorem rngone0 38988

Description: Obsolete theorem, use ringbn0 20483 instead. The base set of a ring is not empty. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
rngone0.1	⊢ 𝐺 = (1st ‘𝑅)
rngone0.2	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
rngone0	⊢ (𝑅 ∈ RingOps → 𝑋 ≠ ∅)

Proof of Theorem rngone0

Step	Hyp	Ref	Expression
1		rngone0.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2	1	rngogrpo 38987	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		rngone0.2	. . 3 ⊢ 𝑋 = ran 𝐺
4	3	grpon0 31119	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑋 ≠ ∅)
5	2, 4	syl 18	1 ⊢ (𝑅 ∈ RingOps → 𝑋 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146   ≠ wne 2959  ∅c0 4281  ran crn 5656  ‘cfv 6547  1^st c1st 8009  GrpOpcgr 31106  RingOpscrngo 38971

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-fo 6553  df-fv 6555  df-ov 7433  df-1st 8011  df-2nd 8012  df-grpo 31110  df-ablo 31162  df-rngo 38972

This theorem is used by:  rngoueqz  39017