Theorem 2idlelb 21507

Description: Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl 21527. (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 21505	. 2 ⊢ 𝑇 = (𝐼 ∩ 𝐽)
6	5	elin2 4148	1 ⊢ (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ‘cfv 6527  opp_rcoppr 20527  LIdealclidl 21445  2Idealc2idl 21503

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-iota 6483  df-fun 6529  df-fv 6535  df-2idl 21504

This theorem is used by:  df2idl2rng  21511  2idlelbas  21519  rng2idlsubgsubrng  21523  2idlcpblrng  21526  2idlcpbl  21527  ker2idl  21534