Theorem wepwso 43830

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 8634	. . . . . 6 ⊢ 2o ∈ ω
2		nnord 7876	. . . . . 6 ⊢ (2o ∈ ω → Ord 2o)
3	1, 2	ax-mp 5	. . . . 5 ⊢ Ord 2o
4		ordwe 6377	. . . . 5 ⊢ (Ord 2o → E We 2o)
5		weso 5654	. . . . 5 ⊢ ( E We 2o → E Or 2o)
6	3, 4, 5	mp2b 10	. . . 4 ⊢ E Or 2o
7		eqid 2765	. . . . 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 9520	. . . 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 3478	. . . 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 2765	. . . . 5 ⊢ (𝑎 ∈ (2o ↑m 𝐴) ↦ (◡𝑎 “ {1o})) = (𝑎 ∈ (2o ↑m 𝐴) ↦ (◡𝑎 “ {1o}))
14	12, 7, 13	wepwsolem 43829	. . . 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 7355	. . . 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 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  Vcvv 3457  𝒫 cpw 4564  {csn 4591   class class class wbr 5111  {copab 5175   ↦ cmpt 5194   E cep 5562   Or wor 5570   We wwe 5615  ^◡ccnv 5662   “ cima 5666  Ord word 6363  ‘cfv 6540   Isom wiso 6541  (class class class)co 7419  ωcom 7868  1_oc1o 8452  2_oc2o 8453   ↑_m cmap 8830

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-1o 8459  df-2o 8460  df-map 8832

This theorem is used by:  aomclem1  43841