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Theorem dford3lem2 41394
Description: Lemma for dford3 41395. (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 6404 . . . 4 (Tr 𝑥 → Tr suc 𝑥)
2 vex 3448 . . . . 5 𝑥 ∈ V
32sucid 6400 . . . 4 𝑥 ∈ suc 𝑥
42sucex 7742 . . . . 5 suc 𝑥 ∈ V
5 treq 5231 . . . . . 6 (𝑐 = suc 𝑥 → (Tr 𝑐 ↔ Tr suc 𝑥))
6 eleq2 2823 . . . . . 6 (𝑐 = suc 𝑥 → (𝑥𝑐𝑥 ∈ suc 𝑥))
75, 6anbi12d 632 . . . . 5 (𝑐 = suc 𝑥 → ((Tr 𝑐𝑥𝑐) ↔ (Tr suc 𝑥𝑥 ∈ suc 𝑥)))
84, 7spcev 3564 . . . 4 ((Tr suc 𝑥𝑥 ∈ suc 𝑥) → ∃𝑐(Tr 𝑐𝑥𝑐))
91, 3, 8sylancl 587 . . 3 (Tr 𝑥 → ∃𝑐(Tr 𝑐𝑥𝑐))
109adantr 482 . 2 ((Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦) → ∃𝑐(Tr 𝑐𝑥𝑐))
11 simprl 770 . . . . . 6 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Tr 𝑎)
12 dford3lem1 41393 . . . . . . . . 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 408 . . . . . . 7 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → ∀𝑏𝑎 𝑏 ∈ On)
16 dfss3 3933 . . . . . . 7 (𝑎 ⊆ On ↔ ∀𝑏𝑎 𝑏 ∈ On)
1715, 16sylibr 233 . . . . . 6 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → 𝑎 ⊆ On)
18 ordon 7712 . . . . . . 7 Ord On
1918a1i 11 . . . . . 6 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Ord On)
20 trssord 6335 . . . . . 6 ((Tr 𝑎𝑎 ⊆ On ∧ Ord On) → Ord 𝑎)
2111, 17, 19, 20syl3anc 1372 . . . . 5 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Ord 𝑎)
22 vex 3448 . . . . . 6 𝑎 ∈ V
2322elon 6327 . . . . 5 (𝑎 ∈ On ↔ Ord 𝑎)
2421, 23sylibr 233 . . . 4 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → 𝑎 ∈ On)
2524ex 414 . . 3 (∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → 𝑎 ∈ On))
26 treq 5231 . . . . 5 (𝑎 = 𝑏 → (Tr 𝑎 ↔ Tr 𝑏))
27 raleq 3308 . . . . 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 2817 . . . 4 (𝑎 = 𝑏 → (𝑎 ∈ On ↔ 𝑏 ∈ On))
3028, 29imbi12d 345 . . 3 (𝑎 = 𝑏 → (((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On)))
31 treq 5231 . . . . 5 (𝑎 = 𝑥 → (Tr 𝑎 ↔ Tr 𝑥))
32 raleq 3308 . . . . 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 2817 . . . 4 (𝑎 = 𝑥 → (𝑎 ∈ On ↔ 𝑥 ∈ On))
3533, 34imbi12d 345 . . 3 (𝑎 = 𝑥 → (((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦) → 𝑥 ∈ On)))
3625, 30, 35setindtrs 41392 . 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 397   = wceq 1542  wex 1782  wcel 2107  wral 3061  wss 3911  Tr wtr 5223  Ord word 6317  Oncon0 6318  suc csuc 6320
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673  ax-reg 9533
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-ord 6321  df-on 6322  df-suc 6324
This theorem is referenced by:  dford3  41395
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