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Theorem wepwso 38113
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 7887 . . . . 5 2𝑜 ∈ ω
2 nnord 7236 . . . . 5 (2𝑜 ∈ ω → Ord 2𝑜)
31, 2ax-mp 5 . . . 4 Ord 2𝑜
4 ordwe 5895 . . . 4 (Ord 2𝑜 → E We 2𝑜)
5 weso 5255 . . . 4 ( E We 2𝑜 → E Or 2𝑜)
63, 4, 5mp2b 10 . . 3 E Or 2𝑜
7 eqid 2758 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
87wemapso 8619 . . 3 ((𝐴𝑉𝑅 We 𝐴 ∧ E Or 2𝑜) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2𝑜𝑚 𝐴))
96, 8mp3an3 1560 . 2 ((𝐴𝑉𝑅 We 𝐴) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2𝑜𝑚 𝐴))
10 elex 3350 . . . 4 (𝐴𝑉𝐴 ∈ V)
11 wepwso.t . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑧𝑦 ∧ ¬ 𝑧𝑥) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑤𝑥𝑤𝑦)))}
12 eqid 2758 . . . . 5 (𝑎 ∈ (2𝑜𝑚 𝐴) ↦ (𝑎 “ {1𝑜})) = (𝑎 ∈ (2𝑜𝑚 𝐴) ↦ (𝑎 “ {1𝑜}))
1311, 7, 12wepwsolem 38112 . . . 4 (𝐴 ∈ V → (𝑎 ∈ (2𝑜𝑚 𝐴) ↦ (𝑎 “ {1𝑜})) Isom {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}, 𝑇((2𝑜𝑚 𝐴), 𝒫 𝐴))
14 isoso 6759 . . . 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))
1510, 13, 143syl 18 . . 3 (𝐴𝑉 → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2𝑜𝑚 𝐴) ↔ 𝑇 Or 𝒫 𝐴))
1615adantr 472 . 2 ((𝐴𝑉𝑅 We 𝐴) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))} Or (2𝑜𝑚 𝐴) ↔ 𝑇 Or 𝒫 𝐴))
179, 16mpbid 222 1 ((𝐴𝑉𝑅 We 𝐴) → 𝑇 Or 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1630  wcel 2137  wral 3048  wrex 3049  Vcvv 3338  𝒫 cpw 4300  {csn 4319   class class class wbr 4802  {copab 4862  cmpt 4879   E cep 5176   Or wor 5184   We wwe 5222  ccnv 5263  cima 5267  Ord word 5881  cfv 6047   Isom wiso 6048  (class class class)co 6811  ωcom 7228  1𝑜c1o 7720  2𝑜c2o 7721  𝑚 cmap 8021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-1o 7727  df-2o 7728  df-map 8023
This theorem is referenced by:  aomclem1  38124
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