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Theorem dford3lem2 43039
Description: Lemma for dford3 43040. (Contributed by Stefan O'Rear, 28-Oct-2014.)
Assertion
Ref Expression
dford3lem2 ((Tr 𝑥 ∧ ∀𝑦𝑥 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.
StepHypRef Expression
1 suctr 6470 . . . 4 (Tr 𝑥 → Tr suc 𝑥)
2 vex 3484 . . . . 5 𝑥 ∈ V
32sucid 6466 . . . 4 𝑥 ∈ suc 𝑥
42sucex 7826 . . . . 5 suc 𝑥 ∈ V
5 treq 5267 . . . . . 6 (𝑐 = suc 𝑥 → (Tr 𝑐 ↔ Tr suc 𝑥))
6 eleq2 2830 . . . . . 6 (𝑐 = suc 𝑥 → (𝑥𝑐𝑥 ∈ suc 𝑥))
75, 6anbi12d 632 . . . . 5 (𝑐 = suc 𝑥 → ((Tr 𝑐𝑥𝑐) ↔ (Tr suc 𝑥𝑥 ∈ suc 𝑥)))
84, 7spcev 3606 . . . 4 ((Tr suc 𝑥𝑥 ∈ suc 𝑥) → ∃𝑐(Tr 𝑐𝑥𝑐))
91, 3, 8sylancl 586 . . 3 (Tr 𝑥 → ∃𝑐(Tr 𝑐𝑥𝑐))
109adantr 480 . 2 ((Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦) → ∃𝑐(Tr 𝑐𝑥𝑐))
11 simprl 771 . . . . . 6 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Tr 𝑎)
12 dford3lem1 43038 . . . . . . . . 9 ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → ∀𝑏𝑎 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦))
13 ralim 3086 . . . . . . . . 9 (∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) → (∀𝑏𝑎 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → ∀𝑏𝑎 𝑏 ∈ On))
1412, 13syl5 34 . . . . . . . 8 (∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → ∀𝑏𝑎 𝑏 ∈ On))
1514imp 406 . . . . . . 7 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → ∀𝑏𝑎 𝑏 ∈ On)
16 dfss3 3972 . . . . . . 7 (𝑎 ⊆ On ↔ ∀𝑏𝑎 𝑏 ∈ On)
1715, 16sylibr 234 . . . . . 6 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → 𝑎 ⊆ On)
18 ordon 7797 . . . . . . 7 Ord On
1918a1i 11 . . . . . 6 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Ord On)
20 trssord 6401 . . . . . 6 ((Tr 𝑎𝑎 ⊆ On ∧ Ord On) → Ord 𝑎)
2111, 17, 19, 20syl3anc 1373 . . . . 5 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Ord 𝑎)
22 vex 3484 . . . . . 6 𝑎 ∈ V
2322elon 6393 . . . . 5 (𝑎 ∈ On ↔ Ord 𝑎)
2421, 23sylibr 234 . . . 4 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → 𝑎 ∈ On)
2524ex 412 . . 3 (∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → 𝑎 ∈ On))
26 treq 5267 . . . . 5 (𝑎 = 𝑏 → (Tr 𝑎 ↔ Tr 𝑏))
27 raleq 3323 . . . . 5 (𝑎 = 𝑏 → (∀𝑦𝑎 Tr 𝑦 ↔ ∀𝑦𝑏 Tr 𝑦))
2826, 27anbi12d 632 . . . 4 (𝑎 = 𝑏 → ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) ↔ (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦)))
29 eleq1w 2824 . . . 4 (𝑎 = 𝑏 → (𝑎 ∈ On ↔ 𝑏 ∈ On))
3028, 29imbi12d 344 . . 3 (𝑎 = 𝑏 → (((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On)))
31 treq 5267 . . . . 5 (𝑎 = 𝑥 → (Tr 𝑎 ↔ Tr 𝑥))
32 raleq 3323 . . . . 5 (𝑎 = 𝑥 → (∀𝑦𝑎 Tr 𝑦 ↔ ∀𝑦𝑥 Tr 𝑦))
3331, 32anbi12d 632 . . . 4 (𝑎 = 𝑥 → ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) ↔ (Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦)))
34 eleq1w 2824 . . . 4 (𝑎 = 𝑥 → (𝑎 ∈ On ↔ 𝑥 ∈ On))
3533, 34imbi12d 344 . . 3 (𝑎 = 𝑥 → (((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦) → 𝑥 ∈ On)))
3625, 30, 35setindtrs 43037 . 2 (∃𝑐(Tr 𝑐𝑥𝑐) → ((Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦) → 𝑥 ∈ On))
3710, 36mpcom 38 1 ((Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦) → 𝑥 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  wss 3951  Tr wtr 5259  Ord word 6383  Oncon0 6384  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-reg 9632
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by:  dford3  43040
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