Theorem bj-restreg 37122

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 9614	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
2		eqcom 2743	. . . 4 ⊢ ((𝑥 ∩ 𝐴) = ∅ ↔ ∅ = (𝑥 ∩ 𝐴))
3	2	rexbii 3084	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅ ↔ ∃𝑥 ∈ 𝐴 ∅ = (𝑥 ∩ 𝐴))
4	1, 3	sylib 218	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∅ = (𝑥 ∩ 𝐴))
5		simpl 482	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 𝐴 ∈ 𝑉)
6		elrest 17446	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (∅ ∈ (𝐴 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐴 ∅ = (𝑥 ∩ 𝐴)))
7	5, 6	syldan 591	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (∅ ∈ (𝐴 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐴 ∅ = (𝑥 ∩ 𝐴)))
8	4, 7	mpbird 257	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∅ ∈ (𝐴 ↾t 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∃wrex 3061   ∩ cin 3930  ∅c0 4313  (class class class)co 7410   ↾_t crest 17439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734  ax-reg 9611

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-rest 17441

This theorem is referenced by: (None)