Theorem 2idlelb 21576

Description: Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl 21596. (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 21574	. 2 ⊢ 𝑇 = (𝐼 ∩ 𝐽)
6	5	elin2 4103	1 ⊢ (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ‘cfv 6558  opp_rcoppr 20596  LIdealclidl 21514  2Idealc2idl 21572

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  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-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-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-iota 6512  df-fun 6560  df-fv 6566  df-2idl 21573

This theorem is used by:  df2idl2rng  21580  2idlelbas  21588  rng2idlsubgsubrng  21592  2idlcpblrng  21595  2idlcpbl  21596  ker2idl  21603