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Theorem cofon1 8619
Description: Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐵 and the upper bound of 𝐴 dominates 𝐵, then their upper bounds are equal. Compare with cofcut1 27242 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
Hypotheses
Ref Expression
cofon1.1 (𝜑𝐴 ∈ 𝒫 On)
cofon1.2 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
cofon1.3 (𝜑𝐵 {𝑧 ∈ On ∣ 𝐴𝑧})
Assertion
Ref Expression
cofon1 (𝜑 {𝑧 ∈ On ∣ 𝐴𝑧} = {𝑤 ∈ On ∣ 𝐵𝑤})
Distinct variable groups:   𝑤,𝐴   𝑥,𝐴   𝑧,𝐴   𝑤,𝐵   𝑥,𝐵,𝑦   𝑧,𝐵   𝜑,𝑧   𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑤)   𝐴(𝑦)

Proof of Theorem cofon1
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq2 3971 . . . . 5 (𝑤 = 𝑧 → (𝐵𝑤𝐵𝑧))
21cbvrabv 3418 . . . 4 {𝑤 ∈ On ∣ 𝐵𝑤} = {𝑧 ∈ On ∣ 𝐵𝑧}
3 sseq1 3970 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑥𝑦𝑎𝑦))
43rexbidv 3176 . . . . . . . . . . 11 (𝑥 = 𝑎 → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 𝑎𝑦))
5 cofon1.2 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
65ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
7 simprr 772 . . . . . . . . . . 11 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → 𝑎𝐴)
84, 6, 7rspcdva 3583 . . . . . . . . . 10 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → ∃𝑦𝐵 𝑎𝑦)
9 sseq2 3971 . . . . . . . . . . 11 (𝑦 = 𝑏 → (𝑎𝑦𝑎𝑏))
109cbvrexvw 3227 . . . . . . . . . 10 (∃𝑦𝐵 𝑎𝑦 ↔ ∃𝑏𝐵 𝑎𝑏)
118, 10sylib 217 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → ∃𝑏𝐵 𝑎𝑏)
12 simprl 770 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → 𝐵𝑧)
1312sselda 3945 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝑏𝑧)
14 cofon1.1 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ 𝒫 On)
1514elpwid 4570 . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ On)
1615ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝐴 ⊆ On)
17 simplrr 777 . . . . . . . . . . . . 13 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝑎𝐴)
1816, 17sseldd 3946 . . . . . . . . . . . 12 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝑎 ∈ On)
19 simpllr 775 . . . . . . . . . . . 12 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝑧 ∈ On)
20 ontr2 6365 . . . . . . . . . . . 12 ((𝑎 ∈ On ∧ 𝑧 ∈ On) → ((𝑎𝑏𝑏𝑧) → 𝑎𝑧))
2118, 19, 20syl2anc 585 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → ((𝑎𝑏𝑏𝑧) → 𝑎𝑧))
2213, 21mpan2d 693 . . . . . . . . . 10 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → (𝑎𝑏𝑎𝑧))
2322rexlimdva 3153 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → (∃𝑏𝐵 𝑎𝑏𝑎𝑧))
2411, 23mpd 15 . . . . . . . 8 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → 𝑎𝑧)
2524expr 458 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ 𝐵𝑧) → (𝑎𝐴𝑎𝑧))
2625ssrdv 3951 . . . . . 6 (((𝜑𝑧 ∈ On) ∧ 𝐵𝑧) → 𝐴𝑧)
2726ex 414 . . . . 5 ((𝜑𝑧 ∈ On) → (𝐵𝑧𝐴𝑧))
2827ss2rabdv 4034 . . . 4 (𝜑 → {𝑧 ∈ On ∣ 𝐵𝑧} ⊆ {𝑧 ∈ On ∣ 𝐴𝑧})
292, 28eqsstrid 3993 . . 3 (𝜑 → {𝑤 ∈ On ∣ 𝐵𝑤} ⊆ {𝑧 ∈ On ∣ 𝐴𝑧})
30 intss 4931 . . 3 ({𝑤 ∈ On ∣ 𝐵𝑤} ⊆ {𝑧 ∈ On ∣ 𝐴𝑧} → {𝑧 ∈ On ∣ 𝐴𝑧} ⊆ {𝑤 ∈ On ∣ 𝐵𝑤})
3129, 30syl 17 . 2 (𝜑 {𝑧 ∈ On ∣ 𝐴𝑧} ⊆ {𝑤 ∈ On ∣ 𝐵𝑤})
32 sseq2 3971 . . . 4 (𝑤 = {𝑧 ∈ On ∣ 𝐴𝑧} → (𝐵𝑤𝐵 {𝑧 ∈ On ∣ 𝐴𝑧}))
33 ssorduni 7714 . . . . . . . . 9 (𝐴 ⊆ On → Ord 𝐴)
3415, 33syl 17 . . . . . . . 8 (𝜑 → Ord 𝐴)
35 ordsuc 7749 . . . . . . . 8 (Ord 𝐴 ↔ Ord suc 𝐴)
3634, 35sylib 217 . . . . . . 7 (𝜑 → Ord suc 𝐴)
3714uniexd 7680 . . . . . . . . 9 (𝜑 𝐴 ∈ V)
38 sucexg 7741 . . . . . . . . 9 ( 𝐴 ∈ V → suc 𝐴 ∈ V)
3937, 38syl 17 . . . . . . . 8 (𝜑 → suc 𝐴 ∈ V)
40 elong 6326 . . . . . . . 8 (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
4139, 40syl 17 . . . . . . 7 (𝜑 → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
4236, 41mpbird 257 . . . . . 6 (𝜑 → suc 𝐴 ∈ On)
43 onsucuni 7764 . . . . . . 7 (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
4415, 43syl 17 . . . . . 6 (𝜑𝐴 ⊆ suc 𝐴)
45 sseq2 3971 . . . . . . 7 (𝑧 = suc 𝐴 → (𝐴𝑧𝐴 ⊆ suc 𝐴))
4645rspcev 3582 . . . . . 6 ((suc 𝐴 ∈ On ∧ 𝐴 ⊆ suc 𝐴) → ∃𝑧 ∈ On 𝐴𝑧)
4742, 44, 46syl2anc 585 . . . . 5 (𝜑 → ∃𝑧 ∈ On 𝐴𝑧)
48 onintrab2 7733 . . . . 5 (∃𝑧 ∈ On 𝐴𝑧 {𝑧 ∈ On ∣ 𝐴𝑧} ∈ On)
4947, 48sylib 217 . . . 4 (𝜑 {𝑧 ∈ On ∣ 𝐴𝑧} ∈ On)
50 cofon1.3 . . . 4 (𝜑𝐵 {𝑧 ∈ On ∣ 𝐴𝑧})
5132, 49, 50elrabd 3648 . . 3 (𝜑 {𝑧 ∈ On ∣ 𝐴𝑧} ∈ {𝑤 ∈ On ∣ 𝐵𝑤})
52 intss1 4925 . . 3 ( {𝑧 ∈ On ∣ 𝐴𝑧} ∈ {𝑤 ∈ On ∣ 𝐵𝑤} → {𝑤 ∈ On ∣ 𝐵𝑤} ⊆ {𝑧 ∈ On ∣ 𝐴𝑧})
5351, 52syl 17 . 2 (𝜑 {𝑤 ∈ On ∣ 𝐵𝑤} ⊆ {𝑧 ∈ On ∣ 𝐴𝑧})
5431, 53eqssd 3962 1 (𝜑 {𝑧 ∈ On ∣ 𝐴𝑧} = {𝑤 ∈ On ∣ 𝐵𝑤})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  wrex 3074  {crab 3408  Vcvv 3446  wss 3911  𝒫 cpw 4561   cuni 4866   cint 4908  Ord word 6317  Oncon0 6318  suc csuc 6320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-tr 5224  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-ord 6321  df-on 6322  df-suc 6324
This theorem is referenced by:  cofon2  8620
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