Theorem wepwso 41785

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 8641	. . . . . 6 ⊢ 2o ∈ ω
2		nnord 7863	. . . . . 6 ⊢ (2o ∈ ω → Ord 2o)
3	1, 2	ax-mp 5	. . . . 5 ⊢ Ord 2o
4		ordwe 6378	. . . . 5 ⊢ (Ord 2o → E We 2o)
5		weso 5668	. . . . 5 ⊢ ( E We 2o → E Or 2o)
6	3, 4, 5	mp2b 10	. . . 4 ⊢ E Or 2o
7		eqid 2733	. . . . 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 9546	. . . 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 690	. . 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 483	. 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 3493	. . . 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 2733	. . . . 5 ⊢ (𝑎 ∈ (2o ↑m 𝐴) ↦ (◡𝑎 “ {1o})) = (𝑎 ∈ (2o ↑m 𝐴) ↦ (◡𝑎 “ {1o}))
14	12, 7, 13	wepwsolem 41784	. . . 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 7345	. . . 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 18	. . 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 482	. 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 231	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝑇 Or 𝒫 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  Vcvv 3475  𝒫 cpw 4603  {csn 4629   class class class wbr 5149  {copab 5211   ↦ cmpt 5232   E cep 5580   Or wor 5588   We wwe 5631  ^◡ccnv 5676   “ cima 5680  Ord word 6364  ‘cfv 6544   Isom wiso 6545  (class class class)co 7409  ωcom 7855  1_oc1o 8459  2_oc2o 8460   ↑_m cmap 8820

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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-1o 8466  df-2o 8467  df-map 8822

This theorem is referenced by:  aomclem1  41796