Theorem dford3lem2 43933

Description: Lemma for dford3 43934. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Assertion

Ref	Expression
dford3lem2	⊢ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On)

Distinct variable group:   𝑥,𝑦

Proof of Theorem dford3lem2

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		suctr 6448	. . . 4 ⊢ (Tr 𝑥 → Tr suc 𝑥)
2		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
3	2	sucid 6444	. . . 4 ⊢ 𝑥 ∈ suc 𝑥
4	2	sucex 7811	. . . . 5 ⊢ suc 𝑥 ∈ V
5		treq 5219	. . . . . 6 ⊢ (𝑐 = suc 𝑥 → (Tr 𝑐 ↔ Tr suc 𝑥))
6		eleq2 2849	. . . . . 6 ⊢ (𝑐 = suc 𝑥 → (𝑥 ∈ 𝑐 ↔ 𝑥 ∈ suc 𝑥))
7	5, 6	anbi12d 644	. . . . 5 ⊢ (𝑐 = suc 𝑥 → ((Tr 𝑐 ∧ 𝑥 ∈ 𝑐) ↔ (Tr suc 𝑥 ∧ 𝑥 ∈ suc 𝑥)))
8	4, 7	spcev 3560	. . . 4 ⊢ ((Tr suc 𝑥 ∧ 𝑥 ∈ suc 𝑥) → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐))
9	1, 3, 8	sylancl 598	. . 3 ⊢ (Tr 𝑥 → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐))
10	9	adantr 486	. 2 ⊢ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐))
11		simprl 783	. . . . . 6 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → Tr 𝑎)
12		dford3lem1 43932	. . . . . . . . 9 ⊢ ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → ∀𝑏 ∈ 𝑎 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦))
13		ralim 3102	. . . . . . . . 9 ⊢ (∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) → (∀𝑏 ∈ 𝑎 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → ∀𝑏 ∈ 𝑎 𝑏 ∈ On))
14	12, 13	syl5 35	. . . . . . . 8 ⊢ (∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → ∀𝑏 ∈ 𝑎 𝑏 ∈ On))
15	14	imp 412	. . . . . . 7 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → ∀𝑏 ∈ 𝑎 𝑏 ∈ On)
16		dfss3 3920	. . . . . . 7 ⊢ (𝑎 ⊆ On ↔ ∀𝑏 ∈ 𝑎 𝑏 ∈ On)
17	15, 16	sylibr 237	. . . . . 6 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → 𝑎 ⊆ On)
18		ordon 7782	. . . . . . 7 ⊢ Ord On
19	18	a1i 11	. . . . . 6 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → Ord On)
20		trssord 6376	. . . . . 6 ⊢ ((Tr 𝑎 ∧ 𝑎 ⊆ On ∧ Ord On) → Ord 𝑎)
21	11, 17, 19, 20	syl3anc 1398	. . . . 5 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → Ord 𝑎)
22		vex 3454	. . . . . 6 ⊢ 𝑎 ∈ V
23	22	elon 6368	. . . . 5 ⊢ (𝑎 ∈ On ↔ Ord 𝑎)
24	21, 23	sylibr 237	. . . 4 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → 𝑎 ∈ On)
25	24	ex 418	. . 3 ⊢ (∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On))
26		treq 5219	. . . . 5 ⊢ (𝑎 = 𝑏 → (Tr 𝑎 ↔ Tr 𝑏))
27		raleq 3316	. . . . 5 ⊢ (𝑎 = 𝑏 → (∀𝑦 ∈ 𝑎 Tr 𝑦 ↔ ∀𝑦 ∈ 𝑏 Tr 𝑦))
28	26, 27	anbi12d 644	. . . 4 ⊢ (𝑎 = 𝑏 → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) ↔ (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦)))
29		eleq1w 2843	. . . 4 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ On ↔ 𝑏 ∈ On))
30	28, 29	imbi12d 347	. . 3 ⊢ (𝑎 = 𝑏 → (((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On)))
31		treq 5219	. . . . 5 ⊢ (𝑎 = 𝑥 → (Tr 𝑎 ↔ Tr 𝑥))
32		raleq 3316	. . . . 5 ⊢ (𝑎 = 𝑥 → (∀𝑦 ∈ 𝑎 Tr 𝑦 ↔ ∀𝑦 ∈ 𝑥 Tr 𝑦))
33	31, 32	anbi12d 644	. . . 4 ⊢ (𝑎 = 𝑥 → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) ↔ (Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦)))
34		eleq1w 2843	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ On ↔ 𝑥 ∈ On))
35	33, 34	imbi12d 347	. . 3 ⊢ (𝑎 = 𝑥 → (((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On)))
36	25, 30, 35	setindtrs 43931	. 2 ⊢ (∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐) → ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On))
37	10, 36	mpcom 39	1 ⊢ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899  Tr wtr 5212  Ord word 6358  Oncon0 6359  suc csuc 6361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398  ax-un 7742  ax-reg 9571

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6362  df-on 6363  df-suc 6365

This theorem is used by:  dford3  43934