Theorem rngone0 38162

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 38161	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		rngone0.2	. . 3 ⊢ 𝑋 = ran 𝐺
4	3	grpon0 30590	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑋 ≠ ∅)
5	2, 4	syl 17	1 ⊢ (𝑅 ∈ RingOps → 𝑋 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∅c0 4287  ran crn 5633  ‘cfv 6500  1^st c1st 7941  GrpOpcgr 30577  RingOpscrngo 38145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-ov 7371  df-1st 7943  df-2nd 7944  df-grpo 30581  df-ablo 30633  df-rngo 38146

This theorem is referenced by:  rngoueqz  38191