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Theorem ordtypelem2 8465
Description: Lemma for ordtype 8478. (Contributed by Mario Carneiro, 24-Jun-2015.)
Hypotheses
Ref Expression
ordtypelem.1 𝐹 = recs(𝐺)
ordtypelem.2 𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}
ordtypelem.3 𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))
ordtypelem.5 𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}
ordtypelem.6 𝑂 = OrdIso(𝑅, 𝐴)
ordtypelem.7 (𝜑𝑅 We 𝐴)
ordtypelem.8 (𝜑𝑅 Se 𝐴)
Assertion
Ref Expression
ordtypelem2 (𝜑 → Ord 𝑇)
Distinct variable groups:   𝑣,𝑢,𝐶   ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧,𝑅   𝐴,,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧   𝑡,𝑂,𝑢,𝑣,𝑥   𝜑,𝑡,𝑥   ,𝐹,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑤,𝑣,𝑢,,𝑗)   𝐶(𝑥,𝑧,𝑤,𝑡,,𝑗)   𝑇(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,,𝑗)   𝐺(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,,𝑗)   𝑂(𝑧,𝑤,,𝑗)

Proof of Theorem ordtypelem2
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 ordtypelem.5 . . . . . . . . . 10 𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}
2 ssrab2 3720 . . . . . . . . . 10 {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡} ⊆ On
31, 2eqsstri 3668 . . . . . . . . 9 𝑇 ⊆ On
43a1i 11 . . . . . . . 8 (𝜑𝑇 ⊆ On)
54sselda 3636 . . . . . . 7 ((𝜑𝑎𝑇) → 𝑎 ∈ On)
6 onss 7032 . . . . . . 7 (𝑎 ∈ On → 𝑎 ⊆ On)
75, 6syl 17 . . . . . 6 ((𝜑𝑎𝑇) → 𝑎 ⊆ On)
8 eloni 5771 . . . . . . . 8 (𝑎 ∈ On → Ord 𝑎)
95, 8syl 17 . . . . . . 7 ((𝜑𝑎𝑇) → Ord 𝑎)
10 imaeq2 5497 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
1110raleqdv 3174 . . . . . . . . . . 11 (𝑥 = 𝑎 → (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡))
1211rexbidv 3081 . . . . . . . . . 10 (𝑥 = 𝑎 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡))
1312, 1elrab2 3399 . . . . . . . . 9 (𝑎𝑇 ↔ (𝑎 ∈ On ∧ ∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡))
1413simprbi 479 . . . . . . . 8 (𝑎𝑇 → ∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡)
1514adantl 481 . . . . . . 7 ((𝜑𝑎𝑇) → ∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡)
16 ordelss 5777 . . . . . . . . 9 ((Ord 𝑎𝑥𝑎) → 𝑥𝑎)
17 imass2 5536 . . . . . . . . 9 (𝑥𝑎 → (𝐹𝑥) ⊆ (𝐹𝑎))
18 ssralv 3699 . . . . . . . . . 10 ((𝐹𝑥) ⊆ (𝐹𝑎) → (∀𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡 → ∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
1918reximdv 3045 . . . . . . . . 9 ((𝐹𝑥) ⊆ (𝐹𝑎) → (∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡 → ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
2016, 17, 193syl 18 . . . . . . . 8 ((Ord 𝑎𝑥𝑎) → (∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡 → ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
2120ralrimdva 2998 . . . . . . 7 (Ord 𝑎 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡 → ∀𝑥𝑎𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
229, 15, 21sylc 65 . . . . . 6 ((𝜑𝑎𝑇) → ∀𝑥𝑎𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡)
23 ssrab 3713 . . . . . 6 (𝑎 ⊆ {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡} ↔ (𝑎 ⊆ On ∧ ∀𝑥𝑎𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
247, 22, 23sylanbrc 699 . . . . 5 ((𝜑𝑎𝑇) → 𝑎 ⊆ {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡})
2524, 1syl6sseqr 3685 . . . 4 ((𝜑𝑎𝑇) → 𝑎𝑇)
2625ralrimiva 2995 . . 3 (𝜑 → ∀𝑎𝑇 𝑎𝑇)
27 dftr3 4789 . . 3 (Tr 𝑇 ↔ ∀𝑎𝑇 𝑎𝑇)
2826, 27sylibr 224 . 2 (𝜑 → Tr 𝑇)
29 ordon 7024 . . 3 Ord On
30 trssord 5778 . . 3 ((Tr 𝑇𝑇 ⊆ On ∧ Ord On) → Ord 𝑇)
313, 29, 30mp3an23 1456 . 2 (Tr 𝑇 → Ord 𝑇)
3228, 31syl 17 1 (𝜑 → Ord 𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607   class class class wbr 4685  cmpt 4762  Tr wtr 4785   Se wse 5100   We wwe 5101  ran crn 5144  cima 5146  Ord word 5760  Oncon0 5761  crio 6650  recscrecs 7512  OrdIsocoi 8455
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
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-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765
This theorem is referenced by:  ordtypelem5  8468  ordtypelem6  8469  ordtypelem7  8470  ordtypelem8  8471  ordtypelem9  8472
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