Theorem 2idlelb 21281

Description: Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl 21300. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.)

Hypotheses

Ref	Expression
2idlel.i	⊢ 𝐼 = (LIdeal‘𝑅)
2idlel.o	⊢ 𝑂 = (oppr‘𝑅)
2idlel.j	⊢ 𝐽 = (LIdeal‘𝑂)
2idlel.t	⊢ 𝑇 = (2Ideal‘𝑅)

Assertion

Ref	Expression
2idlelb	⊢ (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽))

Proof of Theorem 2idlelb

Step	Hyp	Ref	Expression
1		2idlel.i	. . 3 ⊢ 𝐼 = (LIdeal‘𝑅)
2		2idlel.o	. . 3 ⊢ 𝑂 = (oppr‘𝑅)
3		2idlel.j	. . 3 ⊢ 𝐽 = (LIdeal‘𝑂)
4		2idlel.t	. . 3 ⊢ 𝑇 = (2Ideal‘𝑅)
5	1, 2, 3, 4	2idlval 21279	. 2 ⊢ 𝑇 = (𝐼 ∩ 𝐽)
6	5	elin2 4213	1 ⊢ (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  opp_rcoppr 20350  LIdealclidl 21234  2Idealc2idl 21277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-2idl 21278

This theorem is referenced by:  df2idl2rng  21284  2idlelbas  21292  rng2idlsubgsubrng  21296  2idlcpblrng  21299  2idlcpbl  21300