Theorem bj-restreg 35197

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 9284	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
2		eqcom 2745	. . . 4 ⊢ ((𝑥 ∩ 𝐴) = ∅ ↔ ∅ = (𝑥 ∩ 𝐴))
3	2	rexbii 3177	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅ ↔ ∃𝑥 ∈ 𝐴 ∅ = (𝑥 ∩ 𝐴))
4	1, 3	sylib 217	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∅ = (𝑥 ∩ 𝐴))
5		simpl 482	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → 𝐴 ∈ 𝑉)
6		elrest 17055	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (∅ ∈ (𝐴 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐴 ∅ = (𝑥 ∩ 𝐴)))
7	5, 6	syldan 590	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (∅ ∈ (𝐴 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐴 ∅ = (𝑥 ∩ 𝐴)))
8	4, 7	mpbird 256	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∅ ∈ (𝐴 ↾t 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064   ∩ cin 3882  ∅c0 4253  (class class class)co 7255   ↾_t crest 17048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566  ax-reg 9281

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-rest 17050

This theorem is referenced by: (None)