Theorem dford3lem2 43832

Description: Lemma for dford3 43833. (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 6454	. . . 4 ⊢ (Tr 𝑥 → Tr suc 𝑥)
2		vex 3461	. . . . 5 ⊢ 𝑥 ∈ V
3	2	sucid 6450	. . . 4 ⊢ 𝑥 ∈ suc 𝑥
4	2	sucex 7812	. . . . 5 ⊢ suc 𝑥 ∈ V
5		treq 5227	. . . . . 6 ⊢ (𝑐 = suc 𝑥 → (Tr 𝑐 ↔ Tr suc 𝑥))
6		eleq2 2854	. . . . . 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 3567	. . . 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 43831	. . . . . . . . 9 ⊢ ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → ∀𝑏 ∈ 𝑎 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦))
13		ralim 3107	. . . . . . . . 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 3927	. . . . . . 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 7783	. . . . . . 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 6382	. . . . . 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 3461	. . . . . 6 ⊢ 𝑎 ∈ V
23	22	elon 6374	. . . . 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 5227	. . . . 5 ⊢ (𝑎 = 𝑏 → (Tr 𝑎 ↔ Tr 𝑏))
27		raleq 3322	. . . . 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 2848	. . . 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 5227	. . . . 5 ⊢ (𝑎 = 𝑥 → (Tr 𝑎 ↔ Tr 𝑥))
32		raleq 3322	. . . . 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 2848	. . . 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 43830	. 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 2146  ∀wral 3081   ⊆ wss 3906  Tr wtr 5220  Ord word 6364  Oncon0 6365  suc csuc 6367

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406  ax-un 7743  ax-reg 9562

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 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6368  df-on 6369  df-suc 6371

This theorem is used by:  dford3  43833