Theorem dford3lem2 43455

Description: Lemma for dford3 43456. (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 6411	. . . 4 ⊢ (Tr 𝑥 → Tr suc 𝑥)
2		vex 3433	. . . . 5 ⊢ 𝑥 ∈ V
3	2	sucid 6407	. . . 4 ⊢ 𝑥 ∈ suc 𝑥
4	2	sucex 7760	. . . . 5 ⊢ suc 𝑥 ∈ V
5		treq 5199	. . . . . 6 ⊢ (𝑐 = suc 𝑥 → (Tr 𝑐 ↔ Tr suc 𝑥))
6		eleq2 2825	. . . . . 6 ⊢ (𝑐 = suc 𝑥 → (𝑥 ∈ 𝑐 ↔ 𝑥 ∈ suc 𝑥))
7	5, 6	anbi12d 633	. . . . 5 ⊢ (𝑐 = suc 𝑥 → ((Tr 𝑐 ∧ 𝑥 ∈ 𝑐) ↔ (Tr suc 𝑥 ∧ 𝑥 ∈ suc 𝑥)))
8	4, 7	spcev 3548	. . . 4 ⊢ ((Tr suc 𝑥 ∧ 𝑥 ∈ suc 𝑥) → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐))
9	1, 3, 8	sylancl 587	. . 3 ⊢ (Tr 𝑥 → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐))
10	9	adantr 480	. 2 ⊢ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐))
11		simprl 771	. . . . . 6 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → Tr 𝑎)
12		dford3lem1 43454	. . . . . . . . 9 ⊢ ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → ∀𝑏 ∈ 𝑎 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦))
13		ralim 3077	. . . . . . . . 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 406	. . . . . . 7 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → ∀𝑏 ∈ 𝑎 𝑏 ∈ On)
16		dfss3 3910	. . . . . . 7 ⊢ (𝑎 ⊆ On ↔ ∀𝑏 ∈ 𝑎 𝑏 ∈ On)
17	15, 16	sylibr 234	. . . . . 6 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → 𝑎 ⊆ On)
18		ordon 7731	. . . . . . 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 6340	. . . . . 6 ⊢ ((Tr 𝑎 ∧ 𝑎 ⊆ On ∧ Ord On) → Ord 𝑎)
21	11, 17, 19, 20	syl3anc 1374	. . . . 5 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → Ord 𝑎)
22		vex 3433	. . . . . 6 ⊢ 𝑎 ∈ V
23	22	elon 6332	. . . . 5 ⊢ (𝑎 ∈ On ↔ Ord 𝑎)
24	21, 23	sylibr 234	. . . 4 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → 𝑎 ∈ On)
25	24	ex 412	. . 3 ⊢ (∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On))
26		treq 5199	. . . . 5 ⊢ (𝑎 = 𝑏 → (Tr 𝑎 ↔ Tr 𝑏))
27		raleq 3292	. . . . 5 ⊢ (𝑎 = 𝑏 → (∀𝑦 ∈ 𝑎 Tr 𝑦 ↔ ∀𝑦 ∈ 𝑏 Tr 𝑦))
28	26, 27	anbi12d 633	. . . 4 ⊢ (𝑎 = 𝑏 → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) ↔ (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦)))
29		eleq1w 2819	. . . 4 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ On ↔ 𝑏 ∈ On))
30	28, 29	imbi12d 344	. . 3 ⊢ (𝑎 = 𝑏 → (((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On)))
31		treq 5199	. . . . 5 ⊢ (𝑎 = 𝑥 → (Tr 𝑎 ↔ Tr 𝑥))
32		raleq 3292	. . . . 5 ⊢ (𝑎 = 𝑥 → (∀𝑦 ∈ 𝑎 Tr 𝑦 ↔ ∀𝑦 ∈ 𝑥 Tr 𝑦))
33	31, 32	anbi12d 633	. . . 4 ⊢ (𝑎 = 𝑥 → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) ↔ (Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦)))
34		eleq1w 2819	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ On ↔ 𝑥 ∈ On))
35	33, 34	imbi12d 344	. . 3 ⊢ (𝑎 = 𝑥 → (((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On)))
36	25, 30, 35	setindtrs 43453	. 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 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889  Tr wtr 5192  Ord word 6322  Oncon0 6323  suc csuc 6325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375  ax-un 7689  ax-reg 9507

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327  df-suc 6329

This theorem is referenced by:  dford3  43456