Theorem zfregfr 4558

Description: The epsilon 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 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-frind 4317	. 2 ⊢ ( E Fr 𝐴 ↔ ∀𝑠 FrFor E 𝐴𝑠)
2		bi2.04 247	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠)) ↔ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → 𝑥 ∈ 𝑠)))
3	2	albii 1463	. . . . . 6 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → 𝑥 ∈ 𝑠)))
4		df-ral 2453	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → 𝑥 ∈ 𝑠) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → 𝑥 ∈ 𝑠)))
5	3, 4	bitr4i 186	. . . . 5 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠)) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → 𝑥 ∈ 𝑠))
6		sbim 1946	. . . . . . . . . . 11 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 → [𝑦 / 𝑥]𝑥 ∈ 𝑠))
7		clelsb1 2275	. . . . . . . . . . . 12 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
8		clelsb1 2275	. . . . . . . . . . . 12 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑠 ↔ 𝑦 ∈ 𝑠)
9	7, 8	imbi12i 238	. . . . . . . . . . 11 ⊢ (([𝑦 / 𝑥]𝑥 ∈ 𝐴 → [𝑦 / 𝑥]𝑥 ∈ 𝑠) ↔ (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝑠))
10	6, 9	bitri 183	. . . . . . . . . 10 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) ↔ (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝑠))
11	10	ralbii 2476	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝑠))
12		ralcom3 2637	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝑠) ↔ ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑠))
13	11, 12	bitri 183	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) ↔ ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑠))
14		epel 4277	. . . . . . . . . 10 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
15	14	imbi1i 237	. . . . . . . . 9 ⊢ ((𝑦 E 𝑥 → 𝑦 ∈ 𝑠) ↔ (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑠))
16	15	ralbii 2476	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 (𝑦 E 𝑥 → 𝑦 ∈ 𝑠) ↔ ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑠))
17	13, 16	bitr4i 186	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) ↔ ∀𝑦 ∈ 𝐴 (𝑦 E 𝑥 → 𝑦 ∈ 𝑠))
18	17	imbi1i 237	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → 𝑥 ∈ 𝑠) ↔ (∀𝑦 ∈ 𝐴 (𝑦 E 𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠))
19	18	ralbii 2476	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → 𝑥 ∈ 𝑠) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦 E 𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠))
20	5, 19	bitri 183	. . . 4 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠)) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦 E 𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠))
21		ax-setind 4521	. . . . 5 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠)) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠))
22		dfss2 3136	. . . . 5 ⊢ (𝐴 ⊆ 𝑠 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠))
23	21, 22	sylibr 133	. . . 4 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠)) → 𝐴 ⊆ 𝑠)
24	20, 23	sylbir 134	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦 E 𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → 𝐴 ⊆ 𝑠)
25		df-frfor 4316	. . 3 ⊢ ( FrFor E 𝐴𝑠 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦 E 𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → 𝐴 ⊆ 𝑠))
26	24, 25	mpbir 145	. 2 ⊢ FrFor E 𝐴𝑠
27	1, 26	mpgbir 1446	1 ⊢ E Fr 𝐴

Syntax hints:   → wi 4  ∀wal 1346  [wsb 1755   ∈ wcel 2141  ∀wral 2448   ⊆ wss 3121   class class class wbr 3989   E cep 4272   FrFor wfrfor 4312   Fr wfr 4313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-eprel 4274  df-frfor 4316  df-frind 4317

This theorem is referenced by:  ordfr  4559  wessep  4562  reg3exmidlemwe  4563