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Theorem ordtypelem2 8971
 Description: Lemma for ordtype 8984. (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}
21ssrab3 4032 . . . . . . . . 9 𝑇 ⊆ On
32a1i 11 . . . . . . . 8 (𝜑𝑇 ⊆ On)
43sselda 3942 . . . . . . 7 ((𝜑𝑎𝑇) → 𝑎 ∈ On)
5 onss 7490 . . . . . . 7 (𝑎 ∈ On → 𝑎 ⊆ On)
64, 5syl 17 . . . . . 6 ((𝜑𝑎𝑇) → 𝑎 ⊆ On)
7 eloni 6179 . . . . . . . 8 (𝑎 ∈ On → Ord 𝑎)
84, 7syl 17 . . . . . . 7 ((𝜑𝑎𝑇) → Ord 𝑎)
9 imaeq2 5903 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
109raleqdv 3392 . . . . . . . . . . 11 (𝑥 = 𝑎 → (∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∀𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡))
1110rexbidv 3283 . . . . . . . . . 10 (𝑥 = 𝑎 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡 ↔ ∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡))
1211, 1elrab2 3658 . . . . . . . . 9 (𝑎𝑇 ↔ (𝑎 ∈ On ∧ ∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡))
1312simprbi 500 . . . . . . . 8 (𝑎𝑇 → ∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡)
1413adantl 485 . . . . . . 7 ((𝜑𝑎𝑇) → ∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡)
15 ordelss 6185 . . . . . . . . 9 ((Ord 𝑎𝑥𝑎) → 𝑥𝑎)
16 imass2 5943 . . . . . . . . 9 (𝑥𝑎 → (𝐹𝑥) ⊆ (𝐹𝑎))
17 ssralv 4008 . . . . . . . . . 10 ((𝐹𝑥) ⊆ (𝐹𝑎) → (∀𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡 → ∀𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
1817reximdv 3259 . . . . . . . . 9 ((𝐹𝑥) ⊆ (𝐹𝑎) → (∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡 → ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
1915, 16, 183syl 18 . . . . . . . 8 ((Ord 𝑎𝑥𝑎) → (∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡 → ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
2019ralrimdva 3179 . . . . . . 7 (Ord 𝑎 → (∃𝑡𝐴𝑧 ∈ (𝐹𝑎)𝑧𝑅𝑡 → ∀𝑥𝑎𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
218, 14, 20sylc 65 . . . . . 6 ((𝜑𝑎𝑇) → ∀𝑥𝑎𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡)
22 ssrab 4024 . . . . . 6 (𝑎 ⊆ {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡} ↔ (𝑎 ⊆ On ∧ ∀𝑥𝑎𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡))
236, 21, 22sylanbrc 586 . . . . 5 ((𝜑𝑎𝑇) → 𝑎 ⊆ {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡})
2423, 1sseqtrrdi 3993 . . . 4 ((𝜑𝑎𝑇) → 𝑎𝑇)
2524ralrimiva 3174 . . 3 (𝜑 → ∀𝑎𝑇 𝑎𝑇)
26 dftr3 5152 . . 3 (Tr 𝑇 ↔ ∀𝑎𝑇 𝑎𝑇)
2725, 26sylibr 237 . 2 (𝜑 → Tr 𝑇)
28 ordon 7483 . . 3 Ord On
29 trssord 6186 . . 3 ((Tr 𝑇𝑇 ⊆ On ∧ Ord On) → Ord 𝑇)
302, 28, 29mp3an23 1450 . 2 (Tr 𝑇 → Ord 𝑇)
3127, 30syl 17 1 (𝜑 → Ord 𝑇)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ∀wral 3130  ∃wrex 3131  {crab 3134  Vcvv 3469   ⊆ wss 3908   class class class wbr 5042   ↦ cmpt 5122  Tr wtr 5148   Se wse 5489   We wwe 5490  ran crn 5533   “ cima 5535  Ord word 6168  Oncon0 6169  ℩crio 7097  recscrecs 7994  OrdIsocoi 8961 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-ord 6172  df-on 6173 This theorem is referenced by:  ordtypelem5  8974  ordtypelem6  8975  ordtypelem7  8976  ordtypelem8  8977  ordtypelem9  8978
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