Theorem dfepfr 5059

Description: An alternate way of saying that the epsilon 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 5039	. 2 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅))
2		epel 4988	. . . . . . . . 9 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
3	2	a1i 11	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑥 → (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦))
4	3	rabbiia 3173	. . . . . . 7 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦}
5		dfin5 3563	. . . . . . 7 ⊢ (𝑥 ∩ 𝑦) = {𝑧 ∈ 𝑥 ∣ 𝑧 ∈ 𝑦}
6	4, 5	eqtr4i 2646	. . . . . 6 ⊢ {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = (𝑥 ∩ 𝑦)
7	6	eqeq1i 2626	. . . . 5 ⊢ ({𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅ ↔ (𝑥 ∩ 𝑦) = ∅)
8	7	rexbii 3034	. . . 4 ⊢ (∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅ ↔ ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅)
9	8	imbi2i 326	. . 3 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅) ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅))
10	9	albii 1744	. 2 ⊢ (∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧 E 𝑦} = ∅) ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅))
11	1, 10	bitri 264	1 ⊢ ( E Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 (𝑥 ∩ 𝑦) = ∅))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1478   = wceq 1480   ≠ wne 2790  ∃wrex 2908  {crab 2911   ∩ cin 3554   ⊆ wss 3555  ∅c0 3891   class class class wbr 4613   E cep 4983   Fr wfr 5030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-eprel 4985  df-fr 5033

This theorem is referenced by:  onfr  5722  zfregfr  8453  onfrALTlem3  38238  onfrALT  38243  onfrALTlem3VD  38603  onfrALTVD  38607