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Theorem cofonr 8672
Description: Inverse cofinality law for ordinals. Contrast with cofcutr 27795 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
Hypotheses
Ref Expression
cofonr.1 (𝜑𝐴 ∈ On)
cofonr.2 (𝜑𝐴 = {𝑥 ∈ On ∣ 𝑋𝑥})
Assertion
Ref Expression
cofonr (𝜑 → ∀𝑦𝐴𝑧𝑋 𝑦𝑧)
Distinct variable groups:   𝑥,𝐴   𝑧,𝐴   𝑥,𝑋   𝑧,𝑋   𝜑,𝑥,𝑦   𝜑,𝑧,𝑦
Allowed substitution hints:   𝐴(𝑦)   𝑋(𝑦)

Proof of Theorem cofonr
StepHypRef Expression
1 cofonr.1 . . . . . . . 8 (𝜑𝐴 ∈ On)
2 onss 7768 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ⊆ On)
31, 2syl 17 . . . . . . 7 (𝜑𝐴 ⊆ On)
43sselda 3977 . . . . . 6 ((𝜑𝑦𝐴) → 𝑦 ∈ On)
5 eloni 6367 . . . . . 6 (𝑦 ∈ On → Ord 𝑦)
6 ordirr 6375 . . . . . 6 (Ord 𝑦 → ¬ 𝑦𝑦)
74, 5, 63syl 18 . . . . 5 ((𝜑𝑦𝐴) → ¬ 𝑦𝑦)
8 cofonr.2 . . . . . . . . 9 (𝜑𝐴 = {𝑥 ∈ On ∣ 𝑋𝑥})
98adantr 480 . . . . . . . 8 ((𝜑𝑦𝐴) → 𝐴 = {𝑥 ∈ On ∣ 𝑋𝑥})
109adantr 480 . . . . . . 7 (((𝜑𝑦𝐴) ∧ 𝑋𝑦) → 𝐴 = {𝑥 ∈ On ∣ 𝑋𝑥})
114adantr 480 . . . . . . . . 9 (((𝜑𝑦𝐴) ∧ 𝑋𝑦) → 𝑦 ∈ On)
12 simpr 484 . . . . . . . . 9 (((𝜑𝑦𝐴) ∧ 𝑋𝑦) → 𝑋𝑦)
13 sseq2 4003 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑋𝑥𝑋𝑦))
1413elrab 3678 . . . . . . . . 9 (𝑦 ∈ {𝑥 ∈ On ∣ 𝑋𝑥} ↔ (𝑦 ∈ On ∧ 𝑋𝑦))
1511, 12, 14sylanbrc 582 . . . . . . . 8 (((𝜑𝑦𝐴) ∧ 𝑋𝑦) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑋𝑥})
16 intss1 4960 . . . . . . . 8 (𝑦 ∈ {𝑥 ∈ On ∣ 𝑋𝑥} → {𝑥 ∈ On ∣ 𝑋𝑥} ⊆ 𝑦)
1715, 16syl 17 . . . . . . 7 (((𝜑𝑦𝐴) ∧ 𝑋𝑦) → {𝑥 ∈ On ∣ 𝑋𝑥} ⊆ 𝑦)
1810, 17eqsstrd 4015 . . . . . 6 (((𝜑𝑦𝐴) ∧ 𝑋𝑦) → 𝐴𝑦)
19 simplr 766 . . . . . 6 (((𝜑𝑦𝐴) ∧ 𝑋𝑦) → 𝑦𝐴)
2018, 19sseldd 3978 . . . . 5 (((𝜑𝑦𝐴) ∧ 𝑋𝑦) → 𝑦𝑦)
217, 20mtand 813 . . . 4 ((𝜑𝑦𝐴) → ¬ 𝑋𝑦)
22 dfss3 3965 . . . 4 (𝑋𝑦 ↔ ∀𝑧𝑋 𝑧𝑦)
2321, 22sylnib 328 . . 3 ((𝜑𝑦𝐴) → ¬ ∀𝑧𝑋 𝑧𝑦)
248, 1eqeltrrd 2828 . . . . . . . . . 10 (𝜑 {𝑥 ∈ On ∣ 𝑋𝑥} ∈ On)
25 onintrab2 7781 . . . . . . . . . 10 (∃𝑥 ∈ On 𝑋𝑥 {𝑥 ∈ On ∣ 𝑋𝑥} ∈ On)
2624, 25sylibr 233 . . . . . . . . 9 (𝜑 → ∃𝑥 ∈ On 𝑋𝑥)
2726adantr 480 . . . . . . . 8 ((𝜑𝑦𝐴) → ∃𝑥 ∈ On 𝑋𝑥)
28 onss 7768 . . . . . . . . . . 11 (𝑥 ∈ On → 𝑥 ⊆ On)
2928adantl 481 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ 𝑥 ∈ On) → 𝑥 ⊆ On)
30 sstr 3985 . . . . . . . . . . 11 ((𝑋𝑥𝑥 ⊆ On) → 𝑋 ⊆ On)
3130expcom 413 . . . . . . . . . 10 (𝑥 ⊆ On → (𝑋𝑥𝑋 ⊆ On))
3229, 31syl 17 . . . . . . . . 9 (((𝜑𝑦𝐴) ∧ 𝑥 ∈ On) → (𝑋𝑥𝑋 ⊆ On))
3332rexlimdva 3149 . . . . . . . 8 ((𝜑𝑦𝐴) → (∃𝑥 ∈ On 𝑋𝑥𝑋 ⊆ On))
3427, 33mpd 15 . . . . . . 7 ((𝜑𝑦𝐴) → 𝑋 ⊆ On)
3534sselda 3977 . . . . . 6 (((𝜑𝑦𝐴) ∧ 𝑧𝑋) → 𝑧 ∈ On)
36 ontri1 6391 . . . . . 6 ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦𝑧 ↔ ¬ 𝑧𝑦))
374, 35, 36syl2an2r 682 . . . . 5 (((𝜑𝑦𝐴) ∧ 𝑧𝑋) → (𝑦𝑧 ↔ ¬ 𝑧𝑦))
3837rexbidva 3170 . . . 4 ((𝜑𝑦𝐴) → (∃𝑧𝑋 𝑦𝑧 ↔ ∃𝑧𝑋 ¬ 𝑧𝑦))
39 rexnal 3094 . . . 4 (∃𝑧𝑋 ¬ 𝑧𝑦 ↔ ¬ ∀𝑧𝑋 𝑧𝑦)
4038, 39bitrdi 287 . . 3 ((𝜑𝑦𝐴) → (∃𝑧𝑋 𝑦𝑧 ↔ ¬ ∀𝑧𝑋 𝑧𝑦))
4123, 40mpbird 257 . 2 ((𝜑𝑦𝐴) → ∃𝑧𝑋 𝑦𝑧)
4241ralrimiva 3140 1 (𝜑 → ∀𝑦𝐴𝑧𝑋 𝑦𝑧)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3055  wrex 3064  {crab 3426  wss 3943   cint 4943  Ord word 6356  Oncon0 6357
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
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-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
This theorem is referenced by:  naddunif  8691
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