Theorem dford3lem2 39631

Description: Lemma for dford3 39632. (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 6276	. . . 4 ⊢ (Tr 𝑥 → Tr suc 𝑥)
2		vex 3499	. . . . 5 ⊢ 𝑥 ∈ V
3	2	sucid 6272	. . . 4 ⊢ 𝑥 ∈ suc 𝑥
4	2	sucex 7528	. . . . 5 ⊢ suc 𝑥 ∈ V
5		treq 5180	. . . . . 6 ⊢ (𝑐 = suc 𝑥 → (Tr 𝑐 ↔ Tr suc 𝑥))
6		eleq2 2903	. . . . . 6 ⊢ (𝑐 = suc 𝑥 → (𝑥 ∈ 𝑐 ↔ 𝑥 ∈ suc 𝑥))
7	5, 6	anbi12d 632	. . . . 5 ⊢ (𝑐 = suc 𝑥 → ((Tr 𝑐 ∧ 𝑥 ∈ 𝑐) ↔ (Tr suc 𝑥 ∧ 𝑥 ∈ suc 𝑥)))
8	4, 7	spcev 3609	. . . 4 ⊢ ((Tr suc 𝑥 ∧ 𝑥 ∈ suc 𝑥) → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐))
9	1, 3, 8	sylancl 588	. . 3 ⊢ (Tr 𝑥 → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐))
10	9	adantr 483	. 2 ⊢ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐))
11		simprl 769	. . . . . 6 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → Tr 𝑎)
12		dford3lem1 39630	. . . . . . . . 9 ⊢ ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → ∀𝑏 ∈ 𝑎 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦))
13		ralim 3164	. . . . . . . . 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 409	. . . . . . 7 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → ∀𝑏 ∈ 𝑎 𝑏 ∈ On)
16		dfss3 3958	. . . . . . 7 ⊢ (𝑎 ⊆ On ↔ ∀𝑏 ∈ 𝑎 𝑏 ∈ On)
17	15, 16	sylibr 236	. . . . . 6 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → 𝑎 ⊆ On)
18		ordon 7500	. . . . . . 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 6210	. . . . . 6 ⊢ ((Tr 𝑎 ∧ 𝑎 ⊆ On ∧ Ord On) → Ord 𝑎)
21	11, 17, 19, 20	syl3anc 1367	. . . . 5 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → Ord 𝑎)
22		vex 3499	. . . . . 6 ⊢ 𝑎 ∈ V
23	22	elon 6202	. . . . 5 ⊢ (𝑎 ∈ On ↔ Ord 𝑎)
24	21, 23	sylibr 236	. . . 4 ⊢ ((∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → 𝑎 ∈ On)
25	24	ex 415	. . 3 ⊢ (∀𝑏 ∈ 𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On))
26		treq 5180	. . . . 5 ⊢ (𝑎 = 𝑏 → (Tr 𝑎 ↔ Tr 𝑏))
27		raleq 3407	. . . . 5 ⊢ (𝑎 = 𝑏 → (∀𝑦 ∈ 𝑎 Tr 𝑦 ↔ ∀𝑦 ∈ 𝑏 Tr 𝑦))
28	26, 27	anbi12d 632	. . . 4 ⊢ (𝑎 = 𝑏 → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) ↔ (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦)))
29		eleq1w 2897	. . . 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 5180	. . . . 5 ⊢ (𝑎 = 𝑥 → (Tr 𝑎 ↔ Tr 𝑥))
32		raleq 3407	. . . . 5 ⊢ (𝑎 = 𝑥 → (∀𝑦 ∈ 𝑎 Tr 𝑦 ↔ ∀𝑦 ∈ 𝑥 Tr 𝑦))
33	31, 32	anbi12d 632	. . . 4 ⊢ (𝑎 = 𝑥 → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) ↔ (Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦)))
34		eleq1w 2897	. . . 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 39629	. 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 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3140   ⊆ wss 3938  Tr wtr 5174  Ord word 6192  Oncon0 6193  suc csuc 6195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463  ax-reg 9058

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-ord 6196  df-on 6197  df-suc 6199

This theorem is referenced by:  dford3  39632