Theorem zfregfr 4640

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 4397	. 2 ⊢ ( E Fr 𝐴 ↔ ∀𝑠 FrFor E 𝐴𝑠)
2		bi2.04 248	. . . . . . 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 1494	. . . . . 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 2491	. . . . . 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 187	. . . . 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 1982	. . . . . . . . . . 11 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) ↔ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 → [𝑦 / 𝑥]𝑥 ∈ 𝑠))
7		clelsb1 2312	. . . . . . . . . . . 12 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
8		clelsb1 2312	. . . . . . . . . . . 12 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑠 ↔ 𝑦 ∈ 𝑠)
9	7, 8	imbi12i 239	. . . . . . . . . . 11 ⊢ (([𝑦 / 𝑥]𝑥 ∈ 𝐴 → [𝑦 / 𝑥]𝑥 ∈ 𝑠) ↔ (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝑠))
10	6, 9	bitri 184	. . . . . . . . . 10 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) ↔ (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝑠))
11	10	ralbii 2514	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝑠))
12		ralcom3 2676	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝑠) ↔ ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑠))
13	11, 12	bitri 184	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) ↔ ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑠))
14		epel 4357	. . . . . . . . . 10 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
15	14	imbi1i 238	. . . . . . . . 9 ⊢ ((𝑦 E 𝑥 → 𝑦 ∈ 𝑠) ↔ (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑠))
16	15	ralbii 2514	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 (𝑦 E 𝑥 → 𝑦 ∈ 𝑠) ↔ ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑠))
17	13, 16	bitr4i 187	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) ↔ ∀𝑦 ∈ 𝐴 (𝑦 E 𝑥 → 𝑦 ∈ 𝑠))
18	17	imbi1i 238	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → 𝑥 ∈ 𝑠) ↔ (∀𝑦 ∈ 𝐴 (𝑦 E 𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠))
19	18	ralbii 2514	. . . . 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 184	. . . 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 4603	. . . . 5 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠)) → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠))
22		ssalel 3189	. . . . 5 ⊢ (𝐴 ⊆ 𝑠 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠))
23	21, 22	sylibr 134	. . . 4 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑠)) → 𝐴 ⊆ 𝑠)
24	20, 23	sylbir 135	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦 E 𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → 𝐴 ⊆ 𝑠)
25		df-frfor 4396	. . 3 ⊢ ( FrFor E 𝐴𝑠 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦 E 𝑥 → 𝑦 ∈ 𝑠) → 𝑥 ∈ 𝑠) → 𝐴 ⊆ 𝑠))
26	24, 25	mpbir 146	. 2 ⊢ FrFor E 𝐴𝑠
27	1, 26	mpgbir 1477	1 ⊢ E Fr 𝐴

Syntax hints:   → wi 4  ∀wal 1371  [wsb 1786   ∈ wcel 2178  ∀wral 2486   ⊆ wss 3174   class class class wbr 4059   E cep 4352   FrFor wfrfor 4392   Fr wfr 4393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-eprel 4354  df-frfor 4396  df-frind 4397

This theorem is referenced by:  ordfr  4641  wessep  4644  reg3exmidlemwe  4645