Theorem dfepfr 5638

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 5615	. 2 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅))
2		epel 5556	. . . . . . . 8 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
3	2	rabbii 3421	. . . . . . 7 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦}
4		dfin5 3934	. . . . . . 7 ⊢ (𝑥 ∩ 𝑦) = {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦}
5	3, 4	eqtr4i 2761	. . . . . 6 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = (𝑥 ∩ 𝑦)
6	5	eqeq1i 2740	. . . . 5 ⊢ ({𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅ ↔ (𝑥 ∩ 𝑦) = ∅)
7	6	rexbii 3083	. . . 4 ⊢ (∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅ ↔ ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)
8	7	imbi2i 336	. . 3 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅))
9	8	albii 1819	. 2 ⊢ (∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅) ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅))
10	1, 9	bitri 275	1 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ≠ wne 2932  ∃wrex 3060  {crab 3415   ∩ cin 3925   ⊆ wss 3926  ∅c0 4308   class class class wbr 5119   E cep 5552   Fr wfr 5603

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-eprel 5553  df-fr 5606

This theorem is referenced by:  onfr  6391  zfregfr  9619  onfrALTlem3  44569  onfrALT  44574  onfrALTlem3VD  44911  onfrALTVD  44915