Theorem onfr 6396

Description: The ordinal class is well-founded. This proof does not require the axiom of regularity. This lemma is used in ordon 7776 (through epweon 7774) 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 5643	. 2 ⊢ ( E Fr On ↔ ∀𝑥((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
2		n0 4333	. . . 4 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
3		ineq2 4194	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑥 ∩ 𝑧) = (𝑥 ∩ 𝑦))
4	3	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 ∩ 𝑧) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
5	4	rspcev 3606	. . . . . . . 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 4217	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
8		ssel2 3958	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
9		eloni 6367	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → Ord 𝑦)
10		ordfr 6372	. . . . . . . . . . 11 ⊢ (Ord 𝑦 → E Fr 𝑦)
11	8, 9, 10	3syl 18	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → E Fr 𝑦)
12		inss2 4218	. . . . . . . . . . 11 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑦
13		vex 3468	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
14	13	inex1 5292	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ∈ V
15	14	epfrc 5644	. . . . . . . . . . 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 4208	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ 𝑦) ∩ 𝑧) = (𝑥 ∩ (𝑦 ∩ 𝑧))
19	8, 9	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
20		elinel2 4182	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑥 ∩ 𝑦) → 𝑧 ∈ 𝑦)
21		ordelss 6373	. . . . . . . . . . . . . . . 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 4203	. . . . . . . . . . . . . . 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 4200	. . . . . . . . . . . . 13 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (𝑥 ∩ (𝑦 ∩ 𝑧)) = (𝑥 ∩ 𝑧))
26	18, 25	eqtrid 2783	. . . . . . . . . . . 12 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ((𝑥 ∩ 𝑦) ∩ 𝑧) = (𝑥 ∩ 𝑧))
27	26	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅ ↔ (𝑥 ∩ 𝑧) = ∅))
28	27	rexbidva 3163	. . . . . . . . . 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 4033	. . . . . . . 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 3020	. . . . . 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 2933  ∃wrex 3061   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313   E cep 5557   Fr wfr 5608  Ord word 6356  Oncon0 6357

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6360  df-on 6361

This theorem is referenced by:  epweon  7774  epweonALT  7775  on2recsfn  8684  on2recsov  8685  on2ind  8686  on3ind  8687  wffr  44953