Theorem zfregfr 9611

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 5635	. 2 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅))
2		vex 3461	. . . . 5 ⊢ 𝑥 ∈ V
3		zfreg 9601	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅)
4	2, 3	mpan 690	. . . 4 ⊢ (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 (𝑦 ∩ 𝑥) = ∅)
5		incom 4182	. . . . . 6 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
6	5	eqeq1i 2739	. . . . 5 ⊢ ((𝑦 ∩ 𝑥) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅)
7	6	rexbii 3082	. . . 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 1798	1 ⊢ E Fr 𝐴

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  ∃wrex 3059  Vcvv 3457   ∩ cin 3923   ⊆ wss 3924  ∅c0 4306   E cep 5549   Fr wfr 5600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pr 5399  ax-reg 9598

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-br 5117  df-opab 5179  df-eprel 5550  df-fr 5603

This theorem is referenced by:  en2lp  9612  dford2  9626  noinfep  9666  zfregs  9738  bnj852  34873  dford5reg  35721  trelpss  44405