Theorem 2idlelb 21210

Description: Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl 21229. (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 21208	. 2 ⊢ 𝑇 = (𝐼 ∩ 𝐽)
6	5	elin2 4144	1 ⊢ (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ‘cfv 6490  opp_rcoppr 20274  LIdealclidl 21163  2Idealc2idl 21206

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 5231  ax-nul 5241  ax-pr 5368

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-2idl 21207

This theorem is referenced by:  df2idl2rng  21213  2idlelbas  21221  rng2idlsubgsubrng  21225  2idlcpblrng  21228  2idlcpbl  21229