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Theorem dford3lem2 37911
Description: Lemma for dford3 37912. (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 5846 . . . 4 (Tr 𝑥 → Tr suc 𝑥)
2 vex 3234 . . . . 5 𝑥 ∈ V
32sucid 5842 . . . 4 𝑥 ∈ suc 𝑥
42sucex 7053 . . . . 5 suc 𝑥 ∈ V
5 treq 4791 . . . . . 6 (𝑐 = suc 𝑥 → (Tr 𝑐 ↔ Tr suc 𝑥))
6 eleq2 2719 . . . . . 6 (𝑐 = suc 𝑥 → (𝑥𝑐𝑥 ∈ suc 𝑥))
75, 6anbi12d 747 . . . . 5 (𝑐 = suc 𝑥 → ((Tr 𝑐𝑥𝑐) ↔ (Tr suc 𝑥𝑥 ∈ suc 𝑥)))
84, 7spcev 3331 . . . 4 ((Tr suc 𝑥𝑥 ∈ suc 𝑥) → ∃𝑐(Tr 𝑐𝑥𝑐))
91, 3, 8sylancl 695 . . 3 (Tr 𝑥 → ∃𝑐(Tr 𝑐𝑥𝑐))
109adantr 480 . 2 ((Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦) → ∃𝑐(Tr 𝑐𝑥𝑐))
11 simprl 809 . . . . . 6 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Tr 𝑎)
12 dford3lem1 37910 . . . . . . . . 9 ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → ∀𝑏𝑎 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦))
13 ralim 2977 . . . . . . . . 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 444 . . . . . . 7 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → ∀𝑏𝑎 𝑏 ∈ On)
16 dfss3 3625 . . . . . . 7 (𝑎 ⊆ On ↔ ∀𝑏𝑎 𝑏 ∈ On)
1715, 16sylibr 224 . . . . . 6 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → 𝑎 ⊆ On)
18 ordon 7024 . . . . . . 7 Ord On
1918a1i 11 . . . . . 6 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Ord On)
20 trssord 5778 . . . . . 6 ((Tr 𝑎𝑎 ⊆ On ∧ Ord On) → Ord 𝑎)
2111, 17, 19, 20syl3anc 1366 . . . . 5 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Ord 𝑎)
22 vex 3234 . . . . . 6 𝑎 ∈ V
2322elon 5770 . . . . 5 (𝑎 ∈ On ↔ Ord 𝑎)
2421, 23sylibr 224 . . . 4 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → 𝑎 ∈ On)
2524ex 449 . . 3 (∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → 𝑎 ∈ On))
26 treq 4791 . . . . 5 (𝑎 = 𝑏 → (Tr 𝑎 ↔ Tr 𝑏))
27 raleq 3168 . . . . 5 (𝑎 = 𝑏 → (∀𝑦𝑎 Tr 𝑦 ↔ ∀𝑦𝑏 Tr 𝑦))
2826, 27anbi12d 747 . . . 4 (𝑎 = 𝑏 → ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) ↔ (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦)))
29 eleq1 2718 . . . 4 (𝑎 = 𝑏 → (𝑎 ∈ On ↔ 𝑏 ∈ On))
3028, 29imbi12d 333 . . 3 (𝑎 = 𝑏 → (((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On)))
31 treq 4791 . . . . 5 (𝑎 = 𝑥 → (Tr 𝑎 ↔ Tr 𝑥))
32 raleq 3168 . . . . 5 (𝑎 = 𝑥 → (∀𝑦𝑎 Tr 𝑦 ↔ ∀𝑦𝑥 Tr 𝑦))
3331, 32anbi12d 747 . . . 4 (𝑎 = 𝑥 → ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) ↔ (Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦)))
34 eleq1 2718 . . . 4 (𝑎 = 𝑥 → (𝑎 ∈ On ↔ 𝑥 ∈ On))
3533, 34imbi12d 333 . . 3 (𝑎 = 𝑥 → (((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦) → 𝑥 ∈ On)))
3625, 30, 35setindtrs 37909 . 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 383   = wceq 1523  wex 1744  wcel 2030  wral 2941  wss 3607  Tr wtr 4785  Ord word 5760  Oncon0 5761  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767
This theorem is referenced by:  dford3  37912
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