Theorem rngone0 38024

Description: The base set of a ring is not empty. (Contributed by FL, 24-Jan-2010.) (New usage 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 38023	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		rngone0.2	. . 3 ⊢ 𝑋 = ran 𝐺
4	3	grpon0 30503	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑋 ≠ ∅)
5	2, 4	syl 17	1 ⊢ (𝑅 ∈ RingOps → 𝑋 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∅c0 4282  ran crn 5622  ‘cfv 6489  1^st c1st 7928  GrpOpcgr 30490  RingOpscrngo 38007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fo 6495  df-fv 6497  df-ov 7358  df-1st 7930  df-2nd 7931  df-grpo 30494  df-ablo 30546  df-rngo 38008

This theorem is referenced by:  rngoueqz  38053