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Theorem cofon2 8620
Description: Cofinality theorem for ordinals. If 𝐴 and 𝐵 are mutually cofinal, then their upper bounds agree. Compare cofcut2 27244 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
Hypotheses
Ref Expression
cofon2.1 (𝜑𝐴 ∈ 𝒫 On)
cofon2.2 (𝜑𝐵 ∈ 𝒫 On)
cofon2.3 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
cofon2.4 (𝜑 → ∀𝑧𝐵𝑤𝐴 𝑧𝑤)
Assertion
Ref Expression
cofon2 (𝜑 {𝑎 ∈ On ∣ 𝐴𝑎} = {𝑏 ∈ On ∣ 𝐵𝑏})
Distinct variable groups:   𝐴,𝑎,𝑏   𝑤,𝐴,𝑧   𝑥,𝐴   𝐵,𝑎,𝑏   𝑥,𝐵,𝑦   𝑧,𝐵   𝜑,𝑎,𝑏   𝑤,𝑏,𝑧   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝐴(𝑦)   𝐵(𝑤)

Proof of Theorem cofon2
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 cofon2.1 . 2 (𝜑𝐴 ∈ 𝒫 On)
2 cofon2.3 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
3 sseq1 3970 . . . . . . . 8 (𝑧 = 𝑏 → (𝑧𝑤𝑏𝑤))
43rexbidv 3176 . . . . . . 7 (𝑧 = 𝑏 → (∃𝑤𝐴 𝑧𝑤 ↔ ∃𝑤𝐴 𝑏𝑤))
5 cofon2.4 . . . . . . . 8 (𝜑 → ∀𝑧𝐵𝑤𝐴 𝑧𝑤)
65adantr 482 . . . . . . 7 ((𝜑𝑏𝐵) → ∀𝑧𝐵𝑤𝐴 𝑧𝑤)
7 simpr 486 . . . . . . 7 ((𝜑𝑏𝐵) → 𝑏𝐵)
84, 6, 7rspcdva 3583 . . . . . 6 ((𝜑𝑏𝐵) → ∃𝑤𝐴 𝑏𝑤)
9 sseq2 3971 . . . . . . 7 (𝑤 = 𝑐 → (𝑏𝑤𝑏𝑐))
109cbvrexvw 3227 . . . . . 6 (∃𝑤𝐴 𝑏𝑤 ↔ ∃𝑐𝐴 𝑏𝑐)
118, 10sylib 217 . . . . 5 ((𝜑𝑏𝐵) → ∃𝑐𝐴 𝑏𝑐)
12 ssintub 4928 . . . . . . . . 9 𝐴 {𝑎 ∈ On ∣ 𝐴𝑎}
1312a1i 11 . . . . . . . 8 ((𝜑𝑏𝐵) → 𝐴 {𝑎 ∈ On ∣ 𝐴𝑎})
1413sselda 3945 . . . . . . 7 (((𝜑𝑏𝐵) ∧ 𝑐𝐴) → 𝑐 {𝑎 ∈ On ∣ 𝐴𝑎})
15 cofon2.2 . . . . . . . . . . 11 (𝜑𝐵 ∈ 𝒫 On)
1615elpwid 4570 . . . . . . . . . 10 (𝜑𝐵 ⊆ On)
1716ad2antrr 725 . . . . . . . . 9 (((𝜑𝑏𝐵) ∧ 𝑐𝐴) → 𝐵 ⊆ On)
18 simplr 768 . . . . . . . . 9 (((𝜑𝑏𝐵) ∧ 𝑐𝐴) → 𝑏𝐵)
1917, 18sseldd 3946 . . . . . . . 8 (((𝜑𝑏𝐵) ∧ 𝑐𝐴) → 𝑏 ∈ On)
201elpwid 4570 . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ On)
21 ssorduni 7714 . . . . . . . . . . . . . 14 (𝐴 ⊆ On → Ord 𝐴)
2220, 21syl 17 . . . . . . . . . . . . 13 (𝜑 → Ord 𝐴)
23 ordsuc 7749 . . . . . . . . . . . . 13 (Ord 𝐴 ↔ Ord suc 𝐴)
2422, 23sylib 217 . . . . . . . . . . . 12 (𝜑 → Ord suc 𝐴)
251uniexd 7680 . . . . . . . . . . . . 13 (𝜑 𝐴 ∈ V)
26 sucexg 7741 . . . . . . . . . . . . 13 ( 𝐴 ∈ V → suc 𝐴 ∈ V)
27 elong 6326 . . . . . . . . . . . . 13 (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
2825, 26, 273syl 18 . . . . . . . . . . . 12 (𝜑 → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
2924, 28mpbird 257 . . . . . . . . . . 11 (𝜑 → suc 𝐴 ∈ On)
30 onsucuni 7764 . . . . . . . . . . . 12 (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
3120, 30syl 17 . . . . . . . . . . 11 (𝜑𝐴 ⊆ suc 𝐴)
32 sseq2 3971 . . . . . . . . . . . 12 (𝑎 = suc 𝐴 → (𝐴𝑎𝐴 ⊆ suc 𝐴))
3332rspcev 3582 . . . . . . . . . . 11 ((suc 𝐴 ∈ On ∧ 𝐴 ⊆ suc 𝐴) → ∃𝑎 ∈ On 𝐴𝑎)
3429, 31, 33syl2anc 585 . . . . . . . . . 10 (𝜑 → ∃𝑎 ∈ On 𝐴𝑎)
35 onintrab2 7733 . . . . . . . . . 10 (∃𝑎 ∈ On 𝐴𝑎 {𝑎 ∈ On ∣ 𝐴𝑎} ∈ On)
3634, 35sylib 217 . . . . . . . . 9 (𝜑 {𝑎 ∈ On ∣ 𝐴𝑎} ∈ On)
3736ad2antrr 725 . . . . . . . 8 (((𝜑𝑏𝐵) ∧ 𝑐𝐴) → {𝑎 ∈ On ∣ 𝐴𝑎} ∈ On)
38 ontr2 6365 . . . . . . . 8 ((𝑏 ∈ On ∧ {𝑎 ∈ On ∣ 𝐴𝑎} ∈ On) → ((𝑏𝑐𝑐 {𝑎 ∈ On ∣ 𝐴𝑎}) → 𝑏 {𝑎 ∈ On ∣ 𝐴𝑎}))
3919, 37, 38syl2anc 585 . . . . . . 7 (((𝜑𝑏𝐵) ∧ 𝑐𝐴) → ((𝑏𝑐𝑐 {𝑎 ∈ On ∣ 𝐴𝑎}) → 𝑏 {𝑎 ∈ On ∣ 𝐴𝑎}))
4014, 39mpan2d 693 . . . . . 6 (((𝜑𝑏𝐵) ∧ 𝑐𝐴) → (𝑏𝑐𝑏 {𝑎 ∈ On ∣ 𝐴𝑎}))
4140rexlimdva 3153 . . . . 5 ((𝜑𝑏𝐵) → (∃𝑐𝐴 𝑏𝑐𝑏 {𝑎 ∈ On ∣ 𝐴𝑎}))
4211, 41mpd 15 . . . 4 ((𝜑𝑏𝐵) → 𝑏 {𝑎 ∈ On ∣ 𝐴𝑎})
4342ex 414 . . 3 (𝜑 → (𝑏𝐵𝑏 {𝑎 ∈ On ∣ 𝐴𝑎}))
4443ssrdv 3951 . 2 (𝜑𝐵 {𝑎 ∈ On ∣ 𝐴𝑎})
451, 2, 44cofon1 8619 1 (𝜑 {𝑎 ∈ On ∣ 𝐴𝑎} = {𝑏 ∈ On ∣ 𝐵𝑏})
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:  naddunif  8638
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