Theorem dfepfr 5605

Description: An alternate way of saying that the membership relation is well-founded. (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 23-Jun-2015.)

Assertion

Ref	Expression
dfepfr	⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅))

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dfepfr

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffr2 5582	. 2 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅))
2		epel 5524	. . . . . . . 8 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
3	2	rabbii 3398	. . . . . . 7 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦}
4		dfin5 3893	. . . . . . 7 ⊢ (𝑥 ∩ 𝑦) = {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦}
5	3, 4	eqtr4i 2767	. . . . . 6 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = (𝑥 ∩ 𝑦)
6	5	eqeq1i 2746	. . . . 5 ⊢ ({𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅ ↔ (𝑥 ∩ 𝑦) = ∅)
7	6	rexbii 3088	. . . 4 ⊢ (∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅ ↔ ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)
8	7	imbi2i 338	. . 3 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅))
9	8	albii 1827	. 2 ⊢ (∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅) ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅))
10	1, 9	bitri 277	1 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397  ∀wal 1546   = wceq 1548   ≠ wne 2936  ∃wrex 3065  {crab 3393   ∩ cin 3884   ⊆ wss 3885  ∅c0 4264   class class class wbr 5075   E cep 5520   Fr wfr 5571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-eprel 5521  df-fr 5574

This theorem is referenced by:  onfr  6353  zfregfr  9520  onfrALTlem3  45003  onfrALT  45008  onfrALTlem3VD  45345  onfrALTVD  45349