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Theorem dford3lem2 42069
Description: Lemma for dford3 42070. (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 6451 . . . 4 (Tr 𝑥 → Tr suc 𝑥)
2 vex 3477 . . . . 5 𝑥 ∈ V
32sucid 6447 . . . 4 𝑥 ∈ suc 𝑥
42sucex 7797 . . . . 5 suc 𝑥 ∈ V
5 treq 5274 . . . . . 6 (𝑐 = suc 𝑥 → (Tr 𝑐 ↔ Tr suc 𝑥))
6 eleq2 2821 . . . . . 6 (𝑐 = suc 𝑥 → (𝑥𝑐𝑥 ∈ suc 𝑥))
75, 6anbi12d 630 . . . . 5 (𝑐 = suc 𝑥 → ((Tr 𝑐𝑥𝑐) ↔ (Tr suc 𝑥𝑥 ∈ suc 𝑥)))
84, 7spcev 3597 . . . 4 ((Tr suc 𝑥𝑥 ∈ suc 𝑥) → ∃𝑐(Tr 𝑐𝑥𝑐))
91, 3, 8sylancl 585 . . 3 (Tr 𝑥 → ∃𝑐(Tr 𝑐𝑥𝑐))
109adantr 480 . 2 ((Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦) → ∃𝑐(Tr 𝑐𝑥𝑐))
11 simprl 768 . . . . . 6 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Tr 𝑎)
12 dford3lem1 42068 . . . . . . . . 9 ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → ∀𝑏𝑎 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦))
13 ralim 3085 . . . . . . . . 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 3971 . . . . . . 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 7767 . . . . . . 7 Ord On
1918a1i 11 . . . . . 6 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Ord On)
20 trssord 6382 . . . . . 6 ((Tr 𝑎𝑎 ⊆ On ∧ Ord On) → Ord 𝑎)
2111, 17, 19, 20syl3anc 1370 . . . . 5 ((∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦)) → Ord 𝑎)
22 vex 3477 . . . . . 6 𝑎 ∈ V
2322elon 6374 . . . . 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 412 . . 3 (∀𝑏𝑎 ((Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) → 𝑎 ∈ On))
26 treq 5274 . . . . 5 (𝑎 = 𝑏 → (Tr 𝑎 ↔ Tr 𝑏))
27 raleq 3321 . . . . 5 (𝑎 = 𝑏 → (∀𝑦𝑎 Tr 𝑦 ↔ ∀𝑦𝑏 Tr 𝑦))
2826, 27anbi12d 630 . . . 4 (𝑎 = 𝑏 → ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) ↔ (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦)))
29 eleq1w 2815 . . . 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 5274 . . . . 5 (𝑎 = 𝑥 → (Tr 𝑎 ↔ Tr 𝑥))
32 raleq 3321 . . . . 5 (𝑎 = 𝑥 → (∀𝑦𝑎 Tr 𝑦 ↔ ∀𝑦𝑥 Tr 𝑦))
3331, 32anbi12d 630 . . . 4 (𝑎 = 𝑥 → ((Tr 𝑎 ∧ ∀𝑦𝑎 Tr 𝑦) ↔ (Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦)))
34 eleq1w 2815 . . . 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 42067 . 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 1780  wcel 2105  wral 3060  wss 3949  Tr wtr 5266  Ord word 6364  Oncon0 6365  suc csuc 6367
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728  ax-reg 9590
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-suc 6371
This theorem is referenced by:  dford3  42070
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