Theorem ordtypelem2 9491

Description: Lemma for ordtype 9504. (Contributed by Mario Carneiro, 24-Jun-2015.)

Hypotheses

Ref	Expression
ordtypelem.1	⊢ 𝐹 = recs(𝐺)
ordtypelem.2	⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}
ordtypelem.3	⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))
ordtypelem.5	⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
ordtypelem.6	⊢ 𝑂 = OrdIso(𝑅, 𝐴)
ordtypelem.7	⊢ (𝜑 → 𝑅 We 𝐴)
ordtypelem.8	⊢ (𝜑 → 𝑅 Se 𝐴)

Assertion

Ref	Expression
ordtypelem2	⊢ (𝜑 → Ord 𝑇)

Distinct variable groups:   𝑣,𝑢,𝐶   ℎ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧,𝑅   𝐴,ℎ,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧   𝑡,𝑂,𝑢,𝑣,𝑥   𝜑,𝑡,𝑥   ℎ,𝐹,𝑗,𝑡,𝑢,𝑣,𝑤,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑧, 𝑤, 𝑣, 𝑢, ℎ, 𝑗)   𝐶(𝑥, 𝑧, 𝑤, 𝑡, ℎ, 𝑗)   𝑇(𝑥, 𝑧, 𝑤, 𝑣, 𝑢, 𝑡, ℎ, 𝑗)   𝐺(𝑥, 𝑧, 𝑤, 𝑣, 𝑢, 𝑡, ℎ, 𝑗)   𝑂(𝑧, 𝑤, ℎ, 𝑗)

Proof of Theorem ordtypelem2

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtypelem.5	. . . . . . . . . 10 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}
2	1	ssrab3 4039	. . . . . . . . 9 ⊢ 𝑇 ⊆ On
3	2	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ On)
4	3	sselda 3940	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ On)
5		onss 7793	. . . . . . 7 ⊢ (𝑎 ∈ On → 𝑎 ⊆ On)
6	4, 5	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ On)
7		eloni 6377	. . . . . . . 8 ⊢ (𝑎 ∈ On → Ord 𝑎)
8	4, 7	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → Ord 𝑎)
9		imaeq2 6063	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝐹 “ 𝑥) = (𝐹 “ 𝑎))
10	9	raleqdv 3326	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡))
11	10	rexbidv 3192	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡 ↔ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡))
12	11, 1	elrab2 3657	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝑇 ↔ (𝑎 ∈ On ∧ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡))
13	12	simprbi 503	. . . . . . . 8 ⊢ (𝑎 ∈ 𝑇 → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡)
14	13	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡)
15		ordelss 6383	. . . . . . . . 9 ⊢ ((Ord 𝑎 ∧ 𝑥 ∈ 𝑎) → 𝑥 ⊆ 𝑎)
16		imass2 6109	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝑎 → (𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑎))
17		ssralv 4009	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑎) → (∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡 → ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
18	17	reximdv 3183	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑎) → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡 → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
19	15, 16, 18	3syl 19	. . . . . . . 8 ⊢ ((Ord 𝑎 ∧ 𝑥 ∈ 𝑎) → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡 → ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
20	19	ralrimdva 3168	. . . . . . 7 ⊢ (Ord 𝑎 → (∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑎)𝑧𝑅𝑡 → ∀𝑥 ∈ 𝑎 ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
21	8, 14, 20	sylc 66	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ∀𝑥 ∈ 𝑎 ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡)
22		ssrab 4028	. . . . . 6 ⊢ (𝑎 ⊆ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} ↔ (𝑎 ⊆ On ∧ ∀𝑥 ∈ 𝑎 ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡))
23	6, 21, 22	sylanbrc 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡})
24	23, 1	sseqtrrdi 3981	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ⊆ 𝑇)
25	24	ralrimiva 3160	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 𝑎 ⊆ 𝑇)
26		dftr3 5228	. . 3 ⊢ (Tr 𝑇 ↔ ∀𝑎 ∈ 𝑇 𝑎 ⊆ 𝑇)
27	25, 26	sylibr 237	. 2 ⊢ (𝜑 → Tr 𝑇)
28		ordon 7785	. . 3 ⊢ Ord On
29		trssord 6384	. . 3 ⊢ ((Tr 𝑇 ∧ 𝑇 ⊆ On ∧ Ord On) → Ord 𝑇)
30	2, 28, 29	mp3an23 1482	. 2 ⊢ (Tr 𝑇 → Ord 𝑇)
31	27, 30	syl 18	1 ⊢ (𝜑 → Ord 𝑇)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092  {crab 3419  Vcvv 3458   ⊆ wss 3908   class class class wbr 5114   ↦ cmpt 5197  Tr wtr 5223   Se wse 5617   We wwe 5618  ran crn 5667   “ cima 5669  Ord word 6366  Oncon0 6367  ℩crio 7379  recscrecs 8366  OrdIsocoi 9481

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-tr 5224  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6370  df-on 6371

This theorem is used by:  ordtypelem5  9494  ordtypelem6  9495  ordtypelem7  9496  ordtypelem8  9497  ordtypelem9  9498