Theorem wepwso 44193

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 𝐴 ∈ 𝑉. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypothesis

Ref	Expression
wepwso.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑧 ∈ 𝑦 ∧ ¬ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦)))}

Assertion

Ref	Expression
wepwso	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 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.

Step	Hyp	Ref	Expression
1		2onn 8665	. . . . . 6 ⊢ 2o ∈ ω
2		nnord 7895	. . . . . 6 ⊢ (2o ∈ ω → Ord 2o)
3	1, 2	ax-mp 5	. . . . 5 ⊢ Ord 2o
4		ordwe 6382	. . . . 5 ⊢ (Ord 2o → E We 2o)
5		weso 5646	. . . . 5 ⊢ ( E We 2o → E Or 2o)
6	3, 4, 5	mp2b 10	. . . 4 ⊢ E Or 2o
7		eqid 2763	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
8	7	wemapso 9558	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ E Or 2o) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} Or (2o ↑m 𝐴))
9	6, 8	mpan2 704	. . 3 ⊢ (𝑅 We 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} Or (2o ↑m 𝐴))
10	9	adantl 487	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} Or (2o ↑m 𝐴))
11		elex 3475	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
12		wepwso.t	. . . . 5 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑧 ∈ 𝑦 ∧ ¬ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦)))}
13		eqid 2763	. . . . 5 ⊢ (𝑎 ∈ (2o ↑m 𝐴) ↦ (◡𝑎 “ {1o})) = (𝑎 ∈ (2o ↑m 𝐴) ↦ (◡𝑎 “ {1o}))
14	12, 7, 13	wepwsolem 44192	. . . 4 ⊢ (𝐴 ∈ V → (𝑎 ∈ (2o ↑m 𝐴) ↦ (◡𝑎 “ {1o})) Isom {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}, 𝑇((2o ↑m 𝐴), 𝒫 𝐴))
15		isoso 7366	. . . 4 ⊢ ((𝑎 ∈ (2o ↑m 𝐴) ↦ (◡𝑎 “ {1o})) Isom {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}, 𝑇((2o ↑m 𝐴), 𝒫 𝐴) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} Or (2o ↑m 𝐴) ↔ 𝑇 Or 𝒫 𝐴))
16	11, 14, 15	3syl 19	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} Or (2o ↑m 𝐴) ↔ 𝑇 Or 𝒫 𝐴))
17	16	adantr 486	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} Or (2o ↑m 𝐴) ↔ 𝑇 Or 𝒫 𝐴))
18	10, 17	mpbid 235	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝑇 Or 𝒫 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∀wral 3080  ∃wrex 3090  Vcvv 3454  𝒫 cpw 4560  {csn 4588   class class class wbr 5106  {copab 5170   ↦ cmpt 5189   E cep 5554   Or wor 5562   We wwe 5607  ^◡ccnv 5654   “ cima 5658  Ord word 6368  ‘cfv 6547   Isom wiso 6548  (class class class)co 7430  ωcom 7887  1_oc1o 8483  2_oc2o 8484   ↑_m cmap 8861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-isom 6556  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-1st 8011  df-2nd 8012  df-1o 8490  df-2o 8491  df-map 8863

This theorem is used by:  aomclem1  44199