Theorem onfr 5680

Description: The ordinal class is well-founded. This lemma is needed for ordon 6874 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 5023	. 2 ⊢ ( E Fr On ↔ ∀𝑥((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
2		n0 3890	. . . 4 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
3		ineq2 3770	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑥 ∩ 𝑧) = (𝑥 ∩ 𝑦))
4	3	eqeq1d 2612	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑥 ∩ 𝑧) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
5	4	rspcev 3282	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ (𝑥 ∩ 𝑦) = ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
6	5	adantll 746	. . . . . . 7 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) = ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
7		inss1 3795	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
8		ssel2 3563	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
9		eloni 5650	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → Ord 𝑦)
10		ordfr 5655	. . . . . . . . . . 11 ⊢ (Ord 𝑦 → E Fr 𝑦)
11	8, 9, 10	3syl 18	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → E Fr 𝑦)
12		inss2 3796	. . . . . . . . . . 11 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑦
13		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
14	13	inex1 4727	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝑦) ∈ V
15	14	epfrc 5024	. . . . . . . . . . 11 ⊢ (( E Fr 𝑦 ∧ (𝑥 ∩ 𝑦) ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅)
16	12, 15	mp3an2 1404	. . . . . . . . . 10 ⊢ (( E Fr 𝑦 ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅)
17	11, 16	sylan 487	. . . . . . . . 9 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥 ∩ 𝑦)((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅)
18		inass 3785	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ 𝑦) ∩ 𝑧) = (𝑥 ∩ (𝑦 ∩ 𝑧))
19	8, 9	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
20		elinel2 3762	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑥 ∩ 𝑦) → 𝑧 ∈ 𝑦)
21		ordelss 5656	. . . . . . . . . . . . . . . 16 ⊢ ((Ord 𝑦 ∧ 𝑧 ∈ 𝑦) → 𝑧 ⊆ 𝑦)
22	19, 20, 21	syl2an 493	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ⊆ 𝑦)
23		sseqin2 3779	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑦 ∩ 𝑧) = 𝑧)
24	22, 23	sylib 207	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (𝑦 ∩ 𝑧) = 𝑧)
25	24	ineq2d 3776	. . . . . . . . . . . . 13 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (𝑥 ∩ (𝑦 ∩ 𝑧)) = (𝑥 ∩ 𝑧))
26	18, 25	syl5eq 2656	. . . . . . . . . . . 12 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ((𝑥 ∩ 𝑦) ∩ 𝑧) = (𝑥 ∩ 𝑧))
27	26	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (((𝑥 ∩ 𝑦) ∩ 𝑧) = ∅ ↔ (𝑥 ∩ 𝑧) = ∅))
28	27	rexbidva 3031	. . . . . . . . . 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 221	. . . . . . . 8 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ (𝑥 ∩ 𝑦)(𝑥 ∩ 𝑧) = ∅)
31		ssrexv 3630	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ⊆ 𝑥 → (∃𝑧 ∈ (𝑥 ∩ 𝑦)(𝑥 ∩ 𝑧) = ∅ → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
32	7, 30, 31	mpsyl 66	. . . . . . 7 ⊢ (((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) ∧ (𝑥 ∩ 𝑦) ≠ ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
33	6, 32	pm2.61dane 2869	. . . . . 6 ⊢ ((𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
34	33	ex 449	. . . . 5 ⊢ (𝑥 ⊆ On → (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
35	34	exlimdv 1848	. . . 4 ⊢ (𝑥 ⊆ On → (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
36	2, 35	syl5bi 231	. . 3 ⊢ (𝑥 ⊆ On → (𝑥 ≠ ∅ → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅))
37	36	imp 444	. 2 ⊢ ((𝑥 ⊆ On ∧ 𝑥 ≠ ∅) → ∃𝑧 ∈ 𝑥 (𝑥 ∩ 𝑧) = ∅)
38	1, 37	mpgbir 1717	1 ⊢ E Fr On

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   E cep 4947   Fr wfr 4994  Ord word 5639  Oncon0 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644

This theorem is referenced by:  ordon  6874