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Theorem wepwso 42087
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 8643 . . . . . 6 2o ∈ ω
2 nnord 7865 . . . . . 6 (2o ∈ ω → Ord 2o)
31, 2ax-mp 5 . . . . 5 Ord 2o
4 ordwe 6376 . . . . 5 (Ord 2o → E We 2o)
5 weso 5666 . . . . 5 ( E We 2o → E Or 2o)
63, 4, 5mp2b 10 . . . 4 E Or 2o
7 eqid 2730 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
87wemapso 9548 . . . 4 ((𝑅 We 𝐴 ∧ E Or 2o) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2om 𝐴))
96, 8mpan2 687 . . 3 (𝑅 We 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2om 𝐴))
109adantl 480 . 2 ((𝐴𝑉𝑅 We 𝐴) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2om 𝐴))
11 elex 3491 . . . 4 (𝐴𝑉𝐴 ∈ V)
12 wepwso.t . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑧𝑦 ∧ ¬ 𝑧𝑥) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑤𝑥𝑤𝑦)))}
13 eqid 2730 . . . . 5 (𝑎 ∈ (2om 𝐴) ↦ (𝑎 “ {1o})) = (𝑎 ∈ (2om 𝐴) ↦ (𝑎 “ {1o}))
1412, 7, 13wepwsolem 42086 . . . 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 7347 . . . 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 479 . 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 394   = wceq 1539  wcel 2104  wral 3059  wrex 3068  Vcvv 3472  𝒫 cpw 4601  {csn 4627   class class class wbr 5147  {copab 5209  cmpt 5230   E cep 5578   Or wor 5586   We wwe 5629  ccnv 5674  cima 5678  Ord word 6362  cfv 6542   Isom wiso 6543  (class class class)co 7411  ωcom 7857  1oc1o 8461  2oc2o 8462  m cmap 8822
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-1o 8468  df-2o 8469  df-map 8824
This theorem is referenced by:  aomclem1  42098
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