Theorem bj-restreg 33052

Description: A reformulation of the axiom of regularity using elementwise intersection. (RK: might have to be placed later since theorems in this section are to be moved early (in the section related to the algebra of sets).) (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
bj-restreg	|- ( ( A e. V /\ A =/= (/) ) -> (/) e. ( A ↾t A ) )

Proof of Theorem bj-restreg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfreg 8500	. . 3 |- ( ( A e. V /\ A =/= (/) ) -> E. x e. A ( x i^i A ) = (/) )
2		eqcom 2629	. . . 4 |- ( ( x i^i A ) = (/) <-> (/) = ( x i^i A ) )
3	2	rexbii 3041	. . 3 |- ( E. x e. A ( x i^i A ) = (/) <-> E. x e. A (/) = ( x i^i A ) )
4	1, 3	sylib 208	. 2 |- ( ( A e. V /\ A =/= (/) ) -> E. x e. A (/) = ( x i^i A ) )
5		simpl 473	. . 3 |- ( ( A e. V /\ A =/= (/) ) -> A e. V )
6		elrest 16088	. . 3 |- ( ( A e. V /\ A e. V ) -> ( (/) e. ( A ↾t A ) <-> E. x e. A (/) = ( x i^i A ) ) )
7	5, 6	syldan 487	. 2 |- ( ( A e. V /\ A =/= (/) ) -> ( (/) e. ( A ↾t A ) <-> E. x e. A (/) = ( x i^i A ) ) )
8	4, 7	mpbird 247	1 |- ( ( A e. V /\ A =/= (/) ) -> (/) e. ( A ↾t A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   i^i cin 3573   (/)c0 3915  (class class class)co 6650   ↾_t crest 16081

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rest 16083

This theorem is referenced by: (None)