Theorem wepwso 43471

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 8578	. . . . . 6 ⊢ 2o ∈ ω
2		nnord 7825	. . . . . 6 ⊢ (2o ∈ ω → Ord 2o)
3	1, 2	ax-mp 5	. . . . 5 ⊢ Ord 2o
4		ordwe 6336	. . . . 5 ⊢ (Ord 2o → E We 2o)
5		weso 5622	. . . . 5 ⊢ ( E We 2o → E Or 2o)
6	3, 4, 5	mp2b 10	. . . 4 ⊢ E Or 2o
7		eqid 2736	. . . . 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 9466	. . . 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 692	. . 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 481	. 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 3450	. . . 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 2736	. . . . 5 ⊢ (𝑎 ∈ (2o ↑m 𝐴) ↦ (◡𝑎 “ {1o})) = (𝑎 ∈ (2o ↑m 𝐴) ↦ (◡𝑎 “ {1o}))
14	12, 7, 13	wepwsolem 43470	. . . 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 7303	. . . 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 480	. 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 232	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝑇 Or 𝒫 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  Vcvv 3429  𝒫 cpw 4541  {csn 4567   class class class wbr 5085  {copab 5147   ↦ cmpt 5166   E cep 5530   Or wor 5538   We wwe 5583  ^◡ccnv 5630   “ cima 5634  Ord word 6322  ‘cfv 6498   Isom wiso 6499  (class class class)co 7367  ωcom 7817  1_oc1o 8398  2_oc2o 8399   ↑_m cmap 8773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-1o 8405  df-2o 8406  df-map 8775

This theorem is referenced by:  aomclem1  43482