Theorem zfregfr 4551

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 4310	. 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 1458	. . . . . 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 2449	. . . . . 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 1941	. . . . . . . . . . 11 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 → [𝑦 / 𝑥]𝑥 ∈ 𝑠))
7		clelsb1 2271	. . . . . . . . . . . 12 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
8		clelsb1 2271	. . . . . . . . . . . 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 2472	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝑠))
12		ralcom3 2633	. . . . . . . . 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 4270	. . . . . . . . . 10 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
15	14	imbi1i 237	. . . . . . . . 9 ⊢ ((𝑦 E 𝑥 → 𝑦 ∈ 𝑠) ↔ (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑠))
16	15	ralbii 2472	. . . . . . . 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 2472	. . . . 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 4514	. . . . 5 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠)) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠))
22		dfss2 3131	. . . . 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 4309	. . 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 1441	1 ⊢ E Fr 𝐴

Syntax hints:   → wi 4  ∀wal 1341  [wsb 1750   ∈ wcel 2136  ∀wral 2444   ⊆ wss 3116   class class class wbr 3982   E cep 4265   FrFor wfrfor 4305   Fr wfr 4306

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-eprel 4267  df-frfor 4309  df-frind 4310

This theorem is referenced by:  ordfr  4552  wessep  4555  reg3exmidlemwe  4556