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Theorem cofon1 8670
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 27791 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 4003 . . . . 5 (𝑤 = 𝑧 → (𝐵𝑤𝐵𝑧))
21cbvrabv 3436 . . . 4 {𝑤 ∈ On ∣ 𝐵𝑤} = {𝑧 ∈ On ∣ 𝐵𝑧}
3 sseq1 4002 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑥𝑦𝑎𝑦))
43rexbidv 3172 . . . . . . . . . . 11 (𝑥 = 𝑎 → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 𝑎𝑦))
5 cofon1.2 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
65ad2antrr 723 . . . . . . . . . . 11 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
7 simprr 770 . . . . . . . . . . 11 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → 𝑎𝐴)
84, 6, 7rspcdva 3607 . . . . . . . . . 10 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → ∃𝑦𝐵 𝑎𝑦)
9 sseq2 4003 . . . . . . . . . . 11 (𝑦 = 𝑏 → (𝑎𝑦𝑎𝑏))
109cbvrexvw 3229 . . . . . . . . . 10 (∃𝑦𝐵 𝑎𝑦 ↔ ∃𝑏𝐵 𝑎𝑏)
118, 10sylib 217 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → ∃𝑏𝐵 𝑎𝑏)
12 simprl 768 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → 𝐵𝑧)
1312sselda 3977 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝑏𝑧)
14 cofon1.1 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ 𝒫 On)
1514elpwid 4606 . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ On)
1615ad3antrrr 727 . . . . . . . . . . . . 13 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝐴 ⊆ On)
17 simplrr 775 . . . . . . . . . . . . 13 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝑎𝐴)
1816, 17sseldd 3978 . . . . . . . . . . . 12 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝑎 ∈ On)
19 simpllr 773 . . . . . . . . . . . 12 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝑧 ∈ On)
20 ontr2 6404 . . . . . . . . . . . 12 ((𝑎 ∈ On ∧ 𝑧 ∈ On) → ((𝑎𝑏𝑏𝑧) → 𝑎𝑧))
2118, 19, 20syl2anc 583 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → ((𝑎𝑏𝑏𝑧) → 𝑎𝑧))
2213, 21mpan2d 691 . . . . . . . . . 10 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → (𝑎𝑏𝑎𝑧))
2322rexlimdva 3149 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → (∃𝑏𝐵 𝑎𝑏𝑎𝑧))
2411, 23mpd 15 . . . . . . . 8 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → 𝑎𝑧)
2524expr 456 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ 𝐵𝑧) → (𝑎𝐴𝑎𝑧))
2625ssrdv 3983 . . . . . 6 (((𝜑𝑧 ∈ On) ∧ 𝐵𝑧) → 𝐴𝑧)
2726ex 412 . . . . 5 ((𝜑𝑧 ∈ On) → (𝐵𝑧𝐴𝑧))
2827ss2rabdv 4068 . . . 4 (𝜑 → {𝑧 ∈ On ∣ 𝐵𝑧} ⊆ {𝑧 ∈ On ∣ 𝐴𝑧})
292, 28eqsstrid 4025 . . 3 (𝜑 → {𝑤 ∈ On ∣ 𝐵𝑤} ⊆ {𝑧 ∈ On ∣ 𝐴𝑧})
30 intss 4966 . . 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 4003 . . . 4 (𝑤 = {𝑧 ∈ On ∣ 𝐴𝑧} → (𝐵𝑤𝐵 {𝑧 ∈ On ∣ 𝐴𝑧}))
33 ssorduni 7762 . . . . . . . . 9 (𝐴 ⊆ On → Ord 𝐴)
3415, 33syl 17 . . . . . . . 8 (𝜑 → Ord 𝐴)
35 ordsuc 7797 . . . . . . . 8 (Ord 𝐴 ↔ Ord suc 𝐴)
3634, 35sylib 217 . . . . . . 7 (𝜑 → Ord suc 𝐴)
3714uniexd 7728 . . . . . . . . 9 (𝜑 𝐴 ∈ V)
38 sucexg 7789 . . . . . . . . 9 ( 𝐴 ∈ V → suc 𝐴 ∈ V)
3937, 38syl 17 . . . . . . . 8 (𝜑 → suc 𝐴 ∈ V)
40 elong 6365 . . . . . . . 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 7812 . . . . . . 7 (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
4415, 43syl 17 . . . . . 6 (𝜑𝐴 ⊆ suc 𝐴)
45 sseq2 4003 . . . . . . 7 (𝑧 = suc 𝐴 → (𝐴𝑧𝐴 ⊆ suc 𝐴))
4645rspcev 3606 . . . . . 6 ((suc 𝐴 ∈ On ∧ 𝐴 ⊆ suc 𝐴) → ∃𝑧 ∈ On 𝐴𝑧)
4742, 44, 46syl2anc 583 . . . . 5 (𝜑 → ∃𝑧 ∈ On 𝐴𝑧)
48 onintrab2 7781 . . . . 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 3680 . . 3 (𝜑 {𝑧 ∈ On ∣ 𝐴𝑧} ∈ {𝑤 ∈ On ∣ 𝐵𝑤})
52 intss1 4960 . . 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 3994 1 (𝜑 {𝑧 ∈ On ∣ 𝐴𝑧} = {𝑤 ∈ On ∣ 𝐵𝑤})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3055  wrex 3064  {crab 3426  Vcvv 3468  wss 3943  𝒫 cpw 4597   cuni 4902   cint 4943  Ord word 6356  Oncon0 6357  suc csuc 6359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-suc 6363
This theorem is referenced by:  cofon2  8671
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