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Theorem dford3lem2 42068
Description: Lemma for dford3 42069. (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 6449 . . . 4 (Tr 𝑥 → Tr suc 𝑥)
2 vex 3476 . . . . 5 𝑥 ∈ V
32sucid 6445 . . . 4 𝑥 ∈ suc 𝑥
42sucex 7796 . . . . 5 suc 𝑥 ∈ V
5 treq 5272 . . . . . 6 (𝑐 = suc 𝑥 → (Tr 𝑐 ↔ Tr suc 𝑥))
6 eleq2 2820 . . . . . 6 (𝑐 = suc 𝑥 → (𝑥𝑐𝑥 ∈ suc 𝑥))
75, 6anbi12d 629 . . . . 5 (𝑐 = suc 𝑥 → ((Tr 𝑐𝑥𝑐) ↔ (Tr suc 𝑥𝑥 ∈ suc 𝑥)))
84, 7spcev 3595 . . . 4 ((Tr suc 𝑥𝑥 ∈ suc 𝑥) → ∃𝑐(Tr 𝑐𝑥𝑐))
91, 3, 8sylancl 584 . . 3 (Tr 𝑥 → ∃𝑐(Tr 𝑐𝑥𝑐))
109adantr 479 . 2 ((Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦) → ∃𝑐(Tr 𝑐𝑥𝑐))
11 simprl 767 . . . . . 6 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Tr 𝑎)
12 dford3lem1 42067 . . . . . . . . 9 ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → ∀𝑏𝑎 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦))
13 ralim 3084 . . . . . . . . 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 405 . . . . . . 7 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → ∀𝑏𝑎 𝑏 ∈ On)
16 dfss3 3969 . . . . . . 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 7766 . . . . . . 7 Ord On
1918a1i 11 . . . . . 6 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Ord On)
20 trssord 6380 . . . . . 6 ((Tr 𝑎𝑎 ⊆ On ∧ Ord On) → Ord 𝑎)
2111, 17, 19, 20syl3anc 1369 . . . . 5 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Ord 𝑎)
22 vex 3476 . . . . . 6 𝑎 ∈ V
2322elon 6372 . . . . 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 411 . . 3 (∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → 𝑎 ∈ On))
26 treq 5272 . . . . 5 (𝑎 = 𝑏 → (Tr 𝑎 ↔ Tr 𝑏))
27 raleq 3320 . . . . 5 (𝑎 = 𝑏 → (∀𝑦𝑎 Tr 𝑦 ↔ ∀𝑦𝑏 Tr 𝑦))
2826, 27anbi12d 629 . . . 4 (𝑎 = 𝑏 → ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) ↔ (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦)))
29 eleq1w 2814 . . . 4 (𝑎 = 𝑏 → (𝑎 ∈ On ↔ 𝑏 ∈ On))
3028, 29imbi12d 343 . . 3 (𝑎 = 𝑏 → (((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On)))
31 treq 5272 . . . . 5 (𝑎 = 𝑥 → (Tr 𝑎 ↔ Tr 𝑥))
32 raleq 3320 . . . . 5 (𝑎 = 𝑥 → (∀𝑦𝑎 Tr 𝑦 ↔ ∀𝑦𝑥 Tr 𝑦))
3331, 32anbi12d 629 . . . 4 (𝑎 = 𝑥 → ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) ↔ (Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦)))
34 eleq1w 2814 . . . 4 (𝑎 = 𝑥 → (𝑎 ∈ On ↔ 𝑥 ∈ On))
3533, 34imbi12d 343 . . 3 (𝑎 = 𝑥 → (((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦) → 𝑥 ∈ On)))
3625, 30, 35setindtrs 42066 . 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 394   = wceq 1539  wex 1779  wcel 2104  wral 3059  wss 3947  Tr wtr 5264  Ord word 6362  Oncon0 6363  suc csuc 6365
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727  ax-reg 9589
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6366  df-on 6367  df-suc 6369
This theorem is referenced by:  dford3  42069
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