Theorem bj-restreg 37666

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 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∅ ∈ (𝐴 ↾t 𝐴))

Proof of Theorem bj-restreg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfreg 9560	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
2		eqcom 2776	. . . 4 ⊢ ((𝑥 ∩ 𝐴) = ∅ ↔ ∅ = (𝑥 ∩ 𝐴))
3	2	rexbii 3118	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅ ↔ ∃𝑥 ∈ 𝐴 ∅ = (𝑥 ∩ 𝐴))
4	1, 3	sylib 221	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∅ = (𝑥 ∩ 𝐴))
5		simpl 487	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 𝐴 ∈ 𝑉)
6		elrest 17482	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (∅ ∈ (𝐴 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐴 ∅ = (𝑥 ∩ 𝐴)))
7	5, 6	syldan 602	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (∅ ∈ (𝐴 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐴 ∅ = (𝑥 ∩ 𝐴)))
8	4, 7	mpbird 260	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∅ ∈ (𝐴 ↾t 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095   ∩ cin 3912  ∅c0 4294  (class class class)co 7413   ↾_t crest 17475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pr 5407  ax-un 7735  ax-reg 9556

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7416  df-oprab 7417  df-mpo 7418  df-rest 17477

This theorem is referenced by: (None)