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Theorem ordtypelem2 8368
Description: Lemma for ordtype 8381. (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 3666 . . . . . . . . . 10 {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡} ⊆ On
31, 2eqsstri 3614 . . . . . . . . 9 𝑇 ⊆ On
43a1i 11 . . . . . . . 8 (𝜑𝑇 ⊆ On)
54sselda 3583 . . . . . . 7 ((𝜑𝑎𝑇) → 𝑎 ∈ On)
6 onss 6937 . . . . . . 7 (𝑎 ∈ On → 𝑎 ⊆ On)
75, 6syl 17 . . . . . 6 ((𝜑𝑎𝑇) → 𝑎 ⊆ On)
8 eloni 5692 . . . . . . . 8 (𝑎 ∈ On → Ord 𝑎)
95, 8syl 17 . . . . . . 7 ((𝜑𝑎𝑇) → Ord 𝑎)
10 imaeq2 5421 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
1110raleqdv 3133 . . . . . . . . . . 11 (𝑥 = 𝑎 → (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡))
1211rexbidv 3045 . . . . . . . . . 10 (𝑥 = 𝑎 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡))
1312, 1elrab2 3348 . . . . . . . . 9 (𝑎𝑇 ↔ (𝑎 ∈ On ∧ ∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡))
1413simprbi 480 . . . . . . . 8 (𝑎𝑇 → ∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡)
1514adantl 482 . . . . . . 7 ((𝜑𝑎𝑇) → ∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡)
16 ordelss 5698 . . . . . . . . 9 ((Ord 𝑎𝑥𝑎) → 𝑥𝑎)
17 imass2 5460 . . . . . . . . 9 (𝑥𝑎 → (𝐹𝑥) ⊆ (𝐹𝑎))
18 ssralv 3645 . . . . . . . . . 10 ((𝐹𝑥) ⊆ (𝐹𝑎) → (∀𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡 → ∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
1918reximdv 3010 . . . . . . . . 9 ((𝐹𝑥) ⊆ (𝐹𝑎) → (∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡 → ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
2016, 17, 193syl 18 . . . . . . . 8 ((Ord 𝑎𝑥𝑎) → (∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡 → ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
2120ralrimdva 2963 . . . . . . 7 (Ord 𝑎 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡 → ∀𝑥𝑎𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
229, 15, 21sylc 65 . . . . . 6 ((𝜑𝑎𝑇) → ∀𝑥𝑎𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡)
23 ssrab 3659 . . . . . 6 (𝑎 ⊆ {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡} ↔ (𝑎 ⊆ On ∧ ∀𝑥𝑎𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
247, 22, 23sylanbrc 697 . . . . 5 ((𝜑𝑎𝑇) → 𝑎 ⊆ {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡})
2524, 1syl6sseqr 3631 . . . 4 ((𝜑𝑎𝑇) → 𝑎𝑇)
2625ralrimiva 2960 . . 3 (𝜑 → ∀𝑎𝑇 𝑎𝑇)
27 dftr3 4716 . . 3 (Tr 𝑇 ↔ ∀𝑎𝑇 𝑎𝑇)
2826, 27sylibr 224 . 2 (𝜑 → Tr 𝑇)
29 ordon 6929 . . 3 Ord On
30 trssord 5699 . . 3 ((Tr 𝑇𝑇 ⊆ On ∧ Ord On) → Ord 𝑇)
313, 29, 30mp3an23 1413 . 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 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  wss 3555   class class class wbr 4613  cmpt 4673  Tr wtr 4712   Se wse 5031   We wwe 5032  ran crn 5075  cima 5077  Ord word 5681  Oncon0 5682  crio 6564  recscrecs 7412  OrdIsocoi 8358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686
This theorem is referenced by:  ordtypelem5  8371  ordtypelem6  8372  ordtypelem7  8373  ordtypelem8  8374  ordtypelem9  8375
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