Theorem 2idlelb 21286

Description: Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl 21305. (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 21284	. 2 ⊢ 𝑇 = (𝐼 ∩ 𝐽)
6	5	elin2 4226	1 ⊢ (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ‘cfv 6573  opp_rcoppr 20359  LIdealclidl 21239  2Idealc2idl 21282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-2idl 21283

This theorem is referenced by:  df2idl2rng  21289  2idlelbas  21297  rng2idlsubgsubrng  21301  2idlcpblrng  21304  2idlcpbl  21305