Theorem eliinid 39294

Description: Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Assertion

Ref	Expression
eliinid	|- ( ( A e. |^|_ x e. B C /\ x e. B ) -> A e. C )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   C( x)

Proof of Theorem eliinid

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( A e. |^|_ x e. B C /\ x e. B ) -> A e. |^|_ x e. B C )
2		eliin 4525	. . . 4 |- ( A e. |^|_ x e. B C -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )
3	2	adantr 481	. . 3 |- ( ( A e. |^|_ x e. B C /\ x e. B ) -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )
4	1, 3	mpbid 222	. 2 |- ( ( A e. |^|_ x e. B C /\ x e. B ) -> A. x e. B A e. C )
5		rspa 2930	. 2 |- ( ( A. x e. B A e. C /\ x e. B ) -> A e. C )
6	4, 5	sylancom 701	1 |- ( ( A e. |^|_ x e. B C /\ x e. B ) -> A e. C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-iin 4523

This theorem is referenced by:  iinssiin  39312  fnlimfvre  39906  smflimlem2  40980  smflimmpt  41016  smfsuplem1  41017  smfsupmpt  41021  smfsupxr  41022  smfinflem  41023  smfinfmpt  41025  smflimsuplem4  41029  smflimsupmpt  41035  smfliminfmpt  41038