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Theorem wepwso 43055
Description: A well-ordering induces a strict ordering on the power set. EDITORIAL: when well-orderings are set like, this can be strengthened to remove 𝐴𝑉. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypothesis
Ref Expression
wepwso.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑧𝑦 ∧ ¬ 𝑧𝑥) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑤𝑥𝑤𝑦)))}
Assertion
Ref Expression
wepwso ((𝐴𝑉𝑅 We 𝐴) → 𝑇 Or 𝒫 𝐴)
Distinct variable groups:   𝑥,𝑅,𝑦,𝑧,𝑤   𝑥,𝐴,𝑦,𝑧,𝑤
Allowed substitution hints:   𝑇(𝑥,𝑦,𝑧,𝑤)   𝑉(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wepwso
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 2onn 8680 . . . . . 6 2o ∈ ω
2 nnord 7895 . . . . . 6 (2o ∈ ω → Ord 2o)
31, 2ax-mp 5 . . . . 5 Ord 2o
4 ordwe 6397 . . . . 5 (Ord 2o → E We 2o)
5 weso 5676 . . . . 5 ( E We 2o → E Or 2o)
63, 4, 5mp2b 10 . . . 4 E Or 2o
7 eqid 2737 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
87wemapso 9591 . . . 4 ((𝑅 We 𝐴 ∧ E Or 2o) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2om 𝐴))
96, 8mpan2 691 . . 3 (𝑅 We 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2om 𝐴))
109adantl 481 . 2 ((𝐴𝑉𝑅 We 𝐴) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2om 𝐴))
11 elex 3501 . . . 4 (𝐴𝑉𝐴 ∈ V)
12 wepwso.t . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑧𝑦 ∧ ¬ 𝑧𝑥) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑤𝑥𝑤𝑦)))}
13 eqid 2737 . . . . 5 (𝑎 ∈ (2om 𝐴) ↦ (𝑎 “ {1o})) = (𝑎 ∈ (2om 𝐴) ↦ (𝑎 “ {1o}))
1412, 7, 13wepwsolem 43054 . . . 4 (𝐴 ∈ V → (𝑎 ∈ (2om 𝐴) ↦ (𝑎 “ {1o})) Isom {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}, 𝑇((2om 𝐴), 𝒫 𝐴))
15 isoso 7368 . . . 4 ((𝑎 ∈ (2om 𝐴) ↦ (𝑎 “ {1o})) Isom {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}, 𝑇((2om 𝐴), 𝒫 𝐴) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2om 𝐴) ↔ 𝑇 Or 𝒫 𝐴))
1611, 14, 153syl 18 . . 3 (𝐴𝑉 → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2om 𝐴) ↔ 𝑇 Or 𝒫 𝐴))
1716adantr 480 . 2 ((𝐴𝑉𝑅 We 𝐴) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2om 𝐴) ↔ 𝑇 Or 𝒫 𝐴))
1810, 17mpbid 232 1 ((𝐴𝑉𝑅 We 𝐴) → 𝑇 Or 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  𝒫 cpw 4600  {csn 4626   class class class wbr 5143  {copab 5205  cmpt 5225   E cep 5583   Or wor 5591   We wwe 5636  ccnv 5684  cima 5688  Ord word 6383  cfv 6561   Isom wiso 6562  (class class class)co 7431  ωcom 7887  1oc1o 8499  2oc2o 8500  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-map 8868
This theorem is referenced by:  aomclem1  43066
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