Theorem onfr 6400

Description: The ordinal class is well-founded. This proof does not require the axiom of regularity. This lemma is used in ordon 7774 (through epweon 7772) in order to eliminate the need for the axiom of regularity. (Contributed by NM, 17-May-1994.)

Assertion

Ref	Expression
onfr	⊢ E Fr On

Proof of Theorem onfr

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfepfr 5644	. 2 ⊢ ( E Fr On ↔ ∀𝑥((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
2		n0 4306	. . . 4 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
3		ineq2 4166	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑥 ∩ 𝑧) = (𝑥 ∩ 𝑦))
4	3	eqeq1d 2764	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 ∩ 𝑧) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
5	4	rspcev 3580	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝑥 ∩ 𝑦) = ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
6	5	adantll 726	. . . . . . 7 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) = ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
7		inss1 4188	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
8		ssel2 3931	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
9		eloni 6370	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → Ord 𝑦)
10		ordfr 6375	. . . . . . . . . . 11 ⊢ (Ord 𝑦 → E Fr 𝑦)
11	8, 9, 10	3syl 19	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → E Fr 𝑦)
12		inss2 4189	. . . . . . . . . . 11 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑦
13		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
14	13	inex1 5285	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ∈ V
15	14	epfrc 5645	. . . . . . . . . . 11 ⊢ (( E Fr 𝑦 ∧ (𝑥 ∩ 𝑦) ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅)
16	12, 15	mp3an2 1477	. . . . . . . . . 10 ⊢ (( E Fr 𝑦 ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅)
17	11, 16	sylan 591	. . . . . . . . 9 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅)
18		inass 4179	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ 𝑦) ∩ 𝑧) = (𝑥 ∩ (𝑦 ∩ 𝑧))
19	8, 9	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
20		elinel2 4154	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑥 ∩ 𝑦) → 𝑧 ∈ 𝑦)
21		ordelss 6376	. . . . . . . . . . . . . . . 16 ⊢ ((Ord 𝑦 ∧ 𝑧 ∈ 𝑦) → 𝑧 ⊆ 𝑦)
22	19, 20, 21	syl2an 607	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ⊆ 𝑦)
23		sseqin2 4175	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑦 ∩ 𝑧) = 𝑧)
24	22, 23	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (𝑦 ∩ 𝑧) = 𝑧)
25	24	ineq2d 4172	. . . . . . . . . . . . 13 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (𝑥 ∩ (𝑦 ∩ 𝑧)) = (𝑥 ∩ 𝑧))
26	18, 25	eqtrid 2809	. . . . . . . . . . . 12 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ((𝑥 ∩ 𝑦) ∩ 𝑧) = (𝑥 ∩ 𝑧))
27	26	eqeq1d 2764	. . . . . . . . . . 11 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅ ↔ (𝑥 ∩ 𝑧) = ∅))
28	27	rexbidva 3186	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → (∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅ ↔ ∃𝑧 ∈ (𝑥 ∩ 𝑦)(𝑥 ∩ 𝑧) = ∅))
29	28	adantr 485	. . . . . . . . 9 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) ≠ ∅) → (∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅ ↔ ∃𝑧 ∈ (𝑥 ∩ 𝑦)(𝑥 ∩ 𝑧) = ∅))
30	17, 29	mpbid 235	. . . . . . . 8 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥 ∩ 𝑦)(𝑥 ∩ 𝑧) = ∅)
31		ssrexv 4006	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ⊆ 𝑥 → (∃𝑧 ∈ (𝑥 ∩ 𝑦)(𝑥 ∩ 𝑧) = ∅ → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
32	7, 30, 31	mpsyl 69	. . . . . . 7 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
33	6, 32	pm2.61dane 3044	. . . . . 6 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
34	33	ex 417	. . . . 5 ⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
35	34	exlimdv 1962	. . . 4 ⊢ (𝑥 ⊆ On → (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
36	2, 35	biimtrid 245	. . 3 ⊢ (𝑥 ⊆ On → (𝑥 ≠ ∅ → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
37	36	imp 411	. 2 ⊢ ((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
38	1, 37	mpgbir 1828	1 ⊢ E Fr On

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3088   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285   E cep 5559   Fr wfr 5610  Ord word 6359  Oncon0 6360

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-tr 5218  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-ord 6363  df-on 6364

This theorem is used by:  epweon  7772  epweonALT  7773  on2recsfn  8651  on2recsov  8652  on2ind  8653  on3ind  8654  wffr  45698