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Theorem wepwso 41399
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 8593 . . . . . 6 2o ∈ ω
2 nnord 7815 . . . . . 6 (2o ∈ ω → Ord 2o)
31, 2ax-mp 5 . . . . 5 Ord 2o
4 ordwe 6335 . . . . 5 (Ord 2o → E We 2o)
5 weso 5629 . . . . 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 9494 . . . 4 ((𝑅 We 𝐴 ∧ E Or 2o) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2om 𝐴))
96, 8mpan2 690 . . 3 (𝑅 We 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2om 𝐴))
109adantl 483 . 2 ((𝐴𝑉𝑅 We 𝐴) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2om 𝐴))
11 elex 3466 . . . 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 41398 . . . 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 7298 . . . 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 482 . 2 ((𝐴𝑉𝑅 We 𝐴) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2om 𝐴) ↔ 𝑇 Or 𝒫 𝐴))
1810, 17mpbid 231 1 ((𝐴𝑉𝑅 We 𝐴) → 𝑇 Or 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  wrex 3074  Vcvv 3448  𝒫 cpw 4565  {csn 4591   class class class wbr 5110  {copab 5172  cmpt 5193   E cep 5541   Or wor 5549   We wwe 5592  ccnv 5637  cima 5641  Ord word 6321  cfv 6501   Isom wiso 6502  (class class class)co 7362  ωcom 7807  1oc1o 8410  2oc2o 8411  m cmap 8772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-1o 8417  df-2o 8418  df-map 8774
This theorem is referenced by:  aomclem1  41410
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