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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 27404 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 4008 . . . . 5 (𝑤 = 𝑧 → (𝐵𝑤𝐵𝑧))
21cbvrabv 3442 . . . 4 {𝑤 ∈ On ∣ 𝐵𝑤} = {𝑧 ∈ On ∣ 𝐵𝑧}
3 sseq1 4007 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑥𝑦𝑎𝑦))
43rexbidv 3178 . . . . . . . . . . 11 (𝑥 = 𝑎 → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 𝑎𝑦))
5 cofon1.2 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
65ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
7 simprr 771 . . . . . . . . . . 11 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → 𝑎𝐴)
84, 6, 7rspcdva 3613 . . . . . . . . . 10 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → ∃𝑦𝐵 𝑎𝑦)
9 sseq2 4008 . . . . . . . . . . 11 (𝑦 = 𝑏 → (𝑎𝑦𝑎𝑏))
109cbvrexvw 3235 . . . . . . . . . 10 (∃𝑦𝐵 𝑎𝑦 ↔ ∃𝑏𝐵 𝑎𝑏)
118, 10sylib 217 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → ∃𝑏𝐵 𝑎𝑏)
12 simprl 769 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → 𝐵𝑧)
1312sselda 3982 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝑏𝑧)
14 cofon1.1 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ 𝒫 On)
1514elpwid 4611 . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ On)
1615ad3antrrr 728 . . . . . . . . . . . . 13 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝐴 ⊆ On)
17 simplrr 776 . . . . . . . . . . . . 13 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝑎𝐴)
1816, 17sseldd 3983 . . . . . . . . . . . 12 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝑎 ∈ On)
19 simpllr 774 . . . . . . . . . . . 12 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → 𝑧 ∈ On)
20 ontr2 6411 . . . . . . . . . . . 12 ((𝑎 ∈ On ∧ 𝑧 ∈ On) → ((𝑎𝑏𝑏𝑧) → 𝑎𝑧))
2118, 19, 20syl2anc 584 . . . . . . . . . . 11 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → ((𝑎𝑏𝑏𝑧) → 𝑎𝑧))
2213, 21mpan2d 692 . . . . . . . . . 10 ((((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) ∧ 𝑏𝐵) → (𝑎𝑏𝑎𝑧))
2322rexlimdva 3155 . . . . . . . . 9 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → (∃𝑏𝐵 𝑎𝑏𝑎𝑧))
2411, 23mpd 15 . . . . . . . 8 (((𝜑𝑧 ∈ On) ∧ (𝐵𝑧𝑎𝐴)) → 𝑎𝑧)
2524expr 457 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ 𝐵𝑧) → (𝑎𝐴𝑎𝑧))
2625ssrdv 3988 . . . . . 6 (((𝜑𝑧 ∈ On) ∧ 𝐵𝑧) → 𝐴𝑧)
2726ex 413 . . . . 5 ((𝜑𝑧 ∈ On) → (𝐵𝑧𝐴𝑧))
2827ss2rabdv 4073 . . . 4 (𝜑 → {𝑧 ∈ On ∣ 𝐵𝑧} ⊆ {𝑧 ∈ On ∣ 𝐴𝑧})
292, 28eqsstrid 4030 . . 3 (𝜑 → {𝑤 ∈ On ∣ 𝐵𝑤} ⊆ {𝑧 ∈ On ∣ 𝐴𝑧})
30 intss 4973 . . 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 4008 . . . 4 (𝑤 = {𝑧 ∈ On ∣ 𝐴𝑧} → (𝐵𝑤𝐵 {𝑧 ∈ On ∣ 𝐴𝑧}))
33 ssorduni 7765 . . . . . . . . 9 (𝐴 ⊆ On → Ord 𝐴)
3415, 33syl 17 . . . . . . . 8 (𝜑 → Ord 𝐴)
35 ordsuc 7800 . . . . . . . 8 (Ord 𝐴 ↔ Ord suc 𝐴)
3634, 35sylib 217 . . . . . . 7 (𝜑 → Ord suc 𝐴)
3714uniexd 7731 . . . . . . . . 9 (𝜑 𝐴 ∈ V)
38 sucexg 7792 . . . . . . . . 9 ( 𝐴 ∈ V → suc 𝐴 ∈ V)
3937, 38syl 17 . . . . . . . 8 (𝜑 → suc 𝐴 ∈ V)
40 elong 6372 . . . . . . . 8 (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
4139, 40syl 17 . . . . . . 7 (𝜑 → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
4236, 41mpbird 256 . . . . . 6 (𝜑 → suc 𝐴 ∈ On)
43 onsucuni 7815 . . . . . . 7 (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
4415, 43syl 17 . . . . . 6 (𝜑𝐴 ⊆ suc 𝐴)
45 sseq2 4008 . . . . . . 7 (𝑧 = suc 𝐴 → (𝐴𝑧𝐴 ⊆ suc 𝐴))
4645rspcev 3612 . . . . . 6 ((suc 𝐴 ∈ On ∧ 𝐴 ⊆ suc 𝐴) → ∃𝑧 ∈ On 𝐴𝑧)
4742, 44, 46syl2anc 584 . . . . 5 (𝜑 → ∃𝑧 ∈ On 𝐴𝑧)
48 onintrab2 7784 . . . . 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 3685 . . 3 (𝜑 {𝑧 ∈ On ∣ 𝐴𝑧} ∈ {𝑤 ∈ On ∣ 𝐵𝑤})
52 intss1 4967 . . 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 3999 1 (𝜑 {𝑧 ∈ On ∣ 𝐴𝑧} = {𝑤 ∈ On ∣ 𝐵𝑤})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  wrex 3070  {crab 3432  Vcvv 3474  wss 3948  𝒫 cpw 4602   cuni 4908   cint 4950  Ord word 6363  Oncon0 6364  suc csuc 6366
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368  df-suc 6370
This theorem is referenced by:  cofon2  8671
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