Theorem eldiftp 3751

Description: Membership in a set with three elements removed. Similar to eldifsn 3836 and eldifpr 3732. (Contributed by David A. Wheeler, 22-Jul-2017.)

Assertion

Ref	Expression
eldiftp	|- ( A e. ( B \ { C , D , E } ) <-> ( A e. B /\ ( A =/= C /\ A =/= D /\ A =/= E ) ) )

Proof of Theorem eldiftp

Step	Hyp	Ref	Expression
1		eldif 3229	. 2 |- ( A e. ( B \ { C , D , E } ) <-> ( A e. B /\ -. A e. { C , D , E } ) )
2		eltpg 3750	. . . . 5 |- ( A e. B -> ( A e. { C , D , E } <-> ( A = C \/ A = D \/ A = E ) ) )
3	2	notbid 677	. . . 4 |- ( A e. B -> ( -. A e. { C , D , E } <-> -. ( A = C \/ A = D \/ A = E ) ) )
4		ne3anior 2508	. . . 4 |- ( ( A =/= C /\ A =/= D /\ A =/= E ) <-> -. ( A = C \/ A = D \/ A = E ) )
5	3, 4	bitr4di 198	. . 3 |- ( A e. B -> ( -. A e. { C , D , E } <-> ( A =/= C /\ A =/= D /\ A =/= E ) ) )
6	5	pm5.32i 458	. 2 |- ( ( A e. B /\ -. A e. { C , D , E } ) <-> ( A e. B /\ ( A =/= C /\ A =/= D /\ A =/= E ) ) )
7	1, 6	bitri 184	1 |- ( A e. ( B \ { C , D , E } ) <-> ( A e. B /\ ( A =/= C /\ A =/= D /\ A =/= E ) ) )

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   \/ w3o 1008   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   \ cdif 3217   {ctp 3707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-un 3224  df-sn 3711  df-pr 3712  df-tp 3713

This theorem is referenced by: (None)