Theorem rngone0 37900

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 37899	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		rngone0.2	. . 3 ⊢ 𝑋 = ran 𝐺
4	3	grpon0 30437	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑋 ≠ ∅)
5	2, 4	syl 17	1 ⊢ (𝑅 ∈ RingOps → 𝑋 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∅c0 4298  ran crn 5641  ‘cfv 6513  1^st c1st 7968  GrpOpcgr 30424  RingOpscrngo 37883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-fo 6519  df-fv 6521  df-ov 7392  df-1st 7970  df-2nd 7971  df-grpo 30428  df-ablo 30480  df-rngo 37884

This theorem is referenced by:  rngoueqz  37929