Theorem onfr 6359

Description: The ordinal class is well-founded. This proof does not require the axiom of regularity. This lemma is used in ordon 7733 (through epweon 7731) 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 5615	. 2 ⊢ ( E Fr On ↔ ∀𝑥((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
2		n0 4312	. . . 4 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
3		ineq2 4173	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑥 ∩ 𝑧) = (𝑥 ∩ 𝑦))
4	3	eqeq1d 2731	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 ∩ 𝑧) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
5	4	rspcev 3585	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝑥 ∩ 𝑦) = ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
6	5	adantll 714	. . . . . . 7 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) = ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
7		inss1 4196	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
8		ssel2 3938	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
9		eloni 6330	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → Ord 𝑦)
10		ordfr 6335	. . . . . . . . . . 11 ⊢ (Ord 𝑦 → E Fr 𝑦)
11	8, 9, 10	3syl 18	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → E Fr 𝑦)
12		inss2 4197	. . . . . . . . . . 11 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑦
13		vex 3448	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
14	13	inex1 5267	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ∈ V
15	14	epfrc 5616	. . . . . . . . . . 11 ⊢ (( E Fr 𝑦 ∧ (𝑥 ∩ 𝑦) ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅)
16	12, 15	mp3an2 1451	. . . . . . . . . 10 ⊢ (( E Fr 𝑦 ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅)
17	11, 16	sylan 580	. . . . . . . . 9 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅)
18		inass 4187	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ 𝑦) ∩ 𝑧) = (𝑥 ∩ (𝑦 ∩ 𝑧))
19	8, 9	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
20		elinel2 4161	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑥 ∩ 𝑦) → 𝑧 ∈ 𝑦)
21		ordelss 6336	. . . . . . . . . . . . . . . 16 ⊢ ((Ord 𝑦 ∧ 𝑧 ∈ 𝑦) → 𝑧 ⊆ 𝑦)
22	19, 20, 21	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ⊆ 𝑦)
23		sseqin2 4182	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑦 ∩ 𝑧) = 𝑧)
24	22, 23	sylib 218	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (𝑦 ∩ 𝑧) = 𝑧)
25	24	ineq2d 4179	. . . . . . . . . . . . 13 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (𝑥 ∩ (𝑦 ∩ 𝑧)) = (𝑥 ∩ 𝑧))
26	18, 25	eqtrid 2776	. . . . . . . . . . . 12 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ((𝑥 ∩ 𝑦) ∩ 𝑧) = (𝑥 ∩ 𝑧))
27	26	eqeq1d 2731	. . . . . . . . . . 11 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅ ↔ (𝑥 ∩ 𝑧) = ∅))
28	27	rexbidva 3155	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → (∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅ ↔ ∃𝑧 ∈ (𝑥 ∩ 𝑦)(𝑥 ∩ 𝑧) = ∅))
29	28	adantr 480	. . . . . . . . 9 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) ≠ ∅) → (∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅ ↔ ∃𝑧 ∈ (𝑥 ∩ 𝑦)(𝑥 ∩ 𝑧) = ∅))
30	17, 29	mpbid 232	. . . . . . . 8 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥 ∩ 𝑦)(𝑥 ∩ 𝑧) = ∅)
31		ssrexv 4013	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ⊆ 𝑥 → (∃𝑧 ∈ (𝑥 ∩ 𝑦)(𝑥 ∩ 𝑧) = ∅ → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
32	7, 30, 31	mpsyl 68	. . . . . . 7 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
33	6, 32	pm2.61dane 3012	. . . . . 6 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
34	33	ex 412	. . . . 5 ⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
35	34	exlimdv 1933	. . . 4 ⊢ (𝑥 ⊆ On → (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
36	2, 35	biimtrid 242	. . 3 ⊢ (𝑥 ⊆ On → (𝑥 ≠ ∅ → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
37	36	imp 406	. 2 ⊢ ((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
38	1, 37	mpgbir 1799	1 ⊢ E Fr On

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292   E cep 5530   Fr wfr 5581  Ord word 6319  Oncon0 6320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6323  df-on 6324

This theorem is referenced by:  epweon  7731  epweonALT  7732  on2recsfn  8608  on2recsov  8609  on2ind  8610  on3ind  8611  wffr  44944