Theorem dfepfr 5669

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 5646	. 2 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅))
2		epel 5587	. . . . . . . 8 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
3	2	rabbii 3442	. . . . . . 7 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦}
4		dfin5 3959	. . . . . . 7 ⊢ (𝑥 ∩ 𝑦) = {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦}
5	3, 4	eqtr4i 2768	. . . . . 6 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = (𝑥 ∩ 𝑦)
6	5	eqeq1i 2742	. . . . 5 ⊢ ({𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅ ↔ (𝑥 ∩ 𝑦) = ∅)
7	6	rexbii 3094	. . . 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 2940  ∃wrex 3070  {crab 3436   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143   E cep 5583   Fr wfr 5634

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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-fr 5637

This theorem is referenced by:  onfr  6423  zfregfr  9645  onfrALTlem3  44564  onfrALT  44569  onfrALTlem3VD  44907  onfrALTVD  44911