Theorem zfregfr 9500

Description: The membership relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)

Assertion

Ref	Expression
zfregfr	⊢ E Fr 𝐴

Proof of Theorem zfregfr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfepfr 5603	. 2 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅))
2		vex 3440	. . . . 5 ⊢ 𝑥 ∈ V
3		zfreg 9488	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅)
4	2, 3	mpan 690	. . . 4 ⊢ (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅)
5		incom 4160	. . . . . 6 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
6	5	eqeq1i 2734	. . . . 5 ⊢ ((𝑦 ∩ 𝑥) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅)
7	6	rexbii 3076	. . . 4 ⊢ (∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅ ↔ ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)
8	4, 7	sylib 218	. . 3 ⊢ (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)
9	8	adantl 481	. 2 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)
10	1, 9	mpgbir 1799	1 ⊢ E Fr 𝐴

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  Vcvv 3436   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284   E cep 5518   Fr wfr 5569

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-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-reg 9484

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-eprel 5519  df-fr 5572

This theorem is referenced by:  elirrvALT  9501  en2lp  9502  dford2  9516  noinfep  9556  zfregs  9628  bnj852  34904  dford5reg  35776  trelpss  44448