Theorem onfr 6355

Description: The ordinal class is well-founded. This proof does not require the axiom of regularity. This lemma is used in ordon 7722 (through epweon 7720) 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 5607	. 2 ⊢ ( E Fr On ↔ ∀𝑥((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
2		n0 4304	. . . 4 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
3		ineq2 4165	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑥 ∩ 𝑧) = (𝑥 ∩ 𝑦))
4	3	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 ∩ 𝑧) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
5	4	rspcev 3575	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝑥 ∩ 𝑦) = ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
6	5	adantll 715	. . . . . . 7 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) = ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
7		inss1 4188	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
8		ssel2 3927	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
9		eloni 6326	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → Ord 𝑦)
10		ordfr 6331	. . . . . . . . . . 11 ⊢ (Ord 𝑦 → E Fr 𝑦)
11	8, 9, 10	3syl 18	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → E Fr 𝑦)
12		inss2 4189	. . . . . . . . . . 11 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑦
13		vex 3443	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
14	13	inex1 5261	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ∈ V
15	14	epfrc 5608	. . . . . . . . . . 11 ⊢ (( E Fr 𝑦 ∧ (𝑥 ∩ 𝑦) ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅)
16	12, 15	mp3an2 1452	. . . . . . . . . 10 ⊢ (( E Fr 𝑦 ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅)
17	11, 16	sylan 581	. . . . . . . . 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 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
20		elinel2 4153	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑥 ∩ 𝑦) → 𝑧 ∈ 𝑦)
21		ordelss 6332	. . . . . . . . . . . . . . . 16 ⊢ ((Ord 𝑦 ∧ 𝑧 ∈ 𝑦) → 𝑧 ⊆ 𝑦)
22	19, 20, 21	syl2an 597	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ⊆ 𝑦)
23		sseqin2 4174	. . . . . . . . . . . . . . 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 4171	. . . . . . . . . . . . 13 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (𝑥 ∩ (𝑦 ∩ 𝑧)) = (𝑥 ∩ 𝑧))
26	18, 25	eqtrid 2782	. . . . . . . . . . . 12 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ((𝑥 ∩ 𝑦) ∩ 𝑧) = (𝑥 ∩ 𝑧))
27	26	eqeq1d 2737	. . . . . . . . . . 11 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅ ↔ (𝑥 ∩ 𝑧) = ∅))
28	27	rexbidva 3157	. . . . . . . . . 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 4002	. . . . . . . 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 3018	. . . . . 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 1935	. . . 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 1801	1 ⊢ E Fr On

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2931  ∃wrex 3059   ∩ cin 3899   ⊆ wss 3900  ∅c0 4284   E cep 5522   Fr wfr 5573  Ord word 6315  Oncon0 6316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-tr 5205  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-ord 6319  df-on 6320

This theorem is referenced by:  epweon  7720  epweonALT  7721  on2recsfn  8595  on2recsov  8596  on2ind  8597  on3ind  8598  wffr  45239