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Theorem wepwso 43032
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 8679 . . . . . 6 2o ∈ ω
2 nnord 7895 . . . . . 6 (2o ∈ ω → Ord 2o)
31, 2ax-mp 5 . . . . 5 Ord 2o
4 ordwe 6399 . . . . 5 (Ord 2o → E We 2o)
5 weso 5680 . . . . 5 ( E We 2o → E Or 2o)
63, 4, 5mp2b 10 . . . 4 E Or 2o
7 eqid 2735 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
87wemapso 9589 . . . 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 3499 . . . 4 (𝐴𝑉𝐴 ∈ V)
12 wepwso.t . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑧𝑦 ∧ ¬ 𝑧𝑥) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑤𝑥𝑤𝑦)))}
13 eqid 2735 . . . . 5 (𝑎 ∈ (2om 𝐴) ↦ (𝑎 “ {1o})) = (𝑎 ∈ (2om 𝐴) ↦ (𝑎 “ {1o}))
1412, 7, 13wepwsolem 43031 . . . 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 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  𝒫 cpw 4605  {csn 4631   class class class wbr 5148  {copab 5210  cmpt 5231   E cep 5588   Or wor 5596   We wwe 5640  ccnv 5688  cima 5692  Ord word 6385  cfv 6563   Isom wiso 6564  (class class class)co 7431  ωcom 7887  1oc1o 8498  2oc2o 8499  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-map 8867
This theorem is referenced by:  aomclem1  43043
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