Theorem dford3lem2 40849

Description: Lemma for dford3 40850. (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 6349	. . . 4 ⊢ (Tr 𝑥 → Tr suc 𝑥)
2		vex 3436	. . . . 5 ⊢ 𝑥 ∈ V
3	2	sucid 6345	. . . 4 ⊢ 𝑥 ∈ suc 𝑥
4	2	sucex 7656	. . . . 5 ⊢ suc 𝑥 ∈ V
5		treq 5197	. . . . . 6 ⊢ (𝑐 = suc 𝑥 → (Tr 𝑐 ↔ Tr suc 𝑥))
6		eleq2 2827	. . . . . 6 ⊢ (𝑐 = suc 𝑥 → (𝑥 ∈ 𝑐 ↔ 𝑥 ∈ suc 𝑥))
7	5, 6	anbi12d 631	. . . . 5 ⊢ (𝑐 = suc 𝑥 → ((Tr 𝑐 ∧ 𝑥 ∈ 𝑐) ↔ (Tr suc 𝑥 ∧ 𝑥 ∈ suc 𝑥)))
8	4, 7	spcev 3545	. . . 4 ⊢ ((Tr suc 𝑥 ∧ 𝑥 ∈ suc 𝑥) → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐))
9	1, 3, 8	sylancl 586	. . 3 ⊢ (Tr 𝑥 → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐))
10	9	adantr 481	. 2 ⊢ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐))
11		simprl 768	. . . . . 6 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → Tr 𝑎)
12		dford3lem1 40848	. . . . . . . . 9 ⊢ ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → ∀𝑏 ∈ 𝑎 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦))
13		ralim 3083	. . . . . . . . 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 34	. . . . . . . 8 ⊢ (∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → ∀𝑏 ∈ 𝑎 𝑏 ∈ On))
15	14	imp 407	. . . . . . 7 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → ∀𝑏 ∈ 𝑎 𝑏 ∈ On)
16		dfss3 3909	. . . . . . 7 ⊢ (𝑎 ⊆ On ↔ ∀𝑏 ∈ 𝑎 𝑏 ∈ On)
17	15, 16	sylibr 233	. . . . . 6 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → 𝑎 ⊆ On)
18		ordon 7627	. . . . . . 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 6283	. . . . . 6 ⊢ ((Tr 𝑎 ∧ 𝑎 ⊆ On ∧ Ord On) → Ord 𝑎)
21	11, 17, 19, 20	syl3anc 1370	. . . . 5 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → Ord 𝑎)
22		vex 3436	. . . . . 6 ⊢ 𝑎 ∈ V
23	22	elon 6275	. . . . 5 ⊢ (𝑎 ∈ On ↔ Ord 𝑎)
24	21, 23	sylibr 233	. . . 4 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → 𝑎 ∈ On)
25	24	ex 413	. . 3 ⊢ (∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On))
26		treq 5197	. . . . 5 ⊢ (𝑎 = 𝑏 → (Tr 𝑎 ↔ Tr 𝑏))
27		raleq 3342	. . . . 5 ⊢ (𝑎 = 𝑏 → (∀𝑦 ∈ 𝑎 Tr 𝑦 ↔ ∀𝑦 ∈ 𝑏 Tr 𝑦))
28	26, 27	anbi12d 631	. . . 4 ⊢ (𝑎 = 𝑏 → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) ↔ (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦)))
29		eleq1w 2821	. . . 4 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ On ↔ 𝑏 ∈ On))
30	28, 29	imbi12d 345	. . 3 ⊢ (𝑎 = 𝑏 → (((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On)))
31		treq 5197	. . . . 5 ⊢ (𝑎 = 𝑥 → (Tr 𝑎 ↔ Tr 𝑥))
32		raleq 3342	. . . . 5 ⊢ (𝑎 = 𝑥 → (∀𝑦 ∈ 𝑎 Tr 𝑦 ↔ ∀𝑦 ∈ 𝑥 Tr 𝑦))
33	31, 32	anbi12d 631	. . . 4 ⊢ (𝑎 = 𝑥 → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) ↔ (Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦)))
34		eleq1w 2821	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ On ↔ 𝑥 ∈ On))
35	33, 34	imbi12d 345	. . 3 ⊢ (𝑎 = 𝑥 → (((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On)))
36	25, 30, 35	setindtrs 40847	. 2 ⊢ (∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐) → ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On))
37	10, 36	mpcom 38	1 ⊢ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887  Tr wtr 5191  Ord word 6265  Oncon0 6266  suc csuc 6268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-reg 9351

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270  df-suc 6272

This theorem is referenced by:  dford3  40850