MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  canthnum Structured version   Visualization version   GIF version

Theorem canthnum 9468
Description: The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 8110. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.)
Assertion
Ref Expression
canthnum (𝐴𝑉𝐴 ≺ (𝒫 𝐴 ∩ dom card))

Proof of Theorem canthnum
Dummy variables 𝑓 𝑎 𝑟 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4848 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 inex1g 4799 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
3 infpwfidom 8848 . . . 4 ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
41, 2, 33syl 18 . . 3 (𝐴𝑉𝐴 ≼ (𝒫 𝐴 ∩ Fin))
5 inex1g 4799 . . . . 5 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ dom card) ∈ V)
61, 5syl 17 . . . 4 (𝐴𝑉 → (𝒫 𝐴 ∩ dom card) ∈ V)
7 finnum 8771 . . . . . 6 (𝑥 ∈ Fin → 𝑥 ∈ dom card)
87ssriv 3605 . . . . 5 Fin ⊆ dom card
9 sslin 3837 . . . . 5 (Fin ⊆ dom card → (𝒫 𝐴 ∩ Fin) ⊆ (𝒫 𝐴 ∩ dom card))
108, 9ax-mp 5 . . . 4 (𝒫 𝐴 ∩ Fin) ⊆ (𝒫 𝐴 ∩ dom card)
11 ssdomg 7998 . . . 4 ((𝒫 𝐴 ∩ dom card) ∈ V → ((𝒫 𝐴 ∩ Fin) ⊆ (𝒫 𝐴 ∩ dom card) → (𝒫 𝐴 ∩ Fin) ≼ (𝒫 𝐴 ∩ dom card)))
126, 10, 11mpisyl 21 . . 3 (𝐴𝑉 → (𝒫 𝐴 ∩ Fin) ≼ (𝒫 𝐴 ∩ dom card))
13 domtr 8006 . . 3 ((𝐴 ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≼ (𝒫 𝐴 ∩ dom card)) → 𝐴 ≼ (𝒫 𝐴 ∩ dom card))
144, 12, 13syl2anc 693 . 2 (𝐴𝑉𝐴 ≼ (𝒫 𝐴 ∩ dom card))
15 eqid 2621 . . . . . . 7 {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝑓‘(𝑟 “ {𝑦})) = 𝑦))} = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝑓‘(𝑟 “ {𝑦})) = 𝑦))}
1615fpwwecbv 9463 . . . . . 6 {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝑓‘(𝑟 “ {𝑦})) = 𝑦))} = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝑓‘(𝑠 “ {𝑧})) = 𝑧))}
17 eqid 2621 . . . . . 6 dom {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝑓‘(𝑟 “ {𝑦})) = 𝑦))} = dom {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝑓‘(𝑟 “ {𝑦})) = 𝑦))}
18 eqid 2621 . . . . . 6 (({⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝑓‘(𝑟 “ {𝑦})) = 𝑦))}‘ dom {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝑓‘(𝑟 “ {𝑦})) = 𝑦))}) “ {(𝑓 dom {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝑓‘(𝑟 “ {𝑦})) = 𝑦))})}) = (({⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝑓‘(𝑟 “ {𝑦})) = 𝑦))}‘ dom {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝑓‘(𝑟 “ {𝑦})) = 𝑦))}) “ {(𝑓 dom {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝑓‘(𝑟 “ {𝑦})) = 𝑦))})})
1916, 17, 18canthnumlem 9467 . . . . 5 (𝐴𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
20 f1of1 6134 . . . . 5 (𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴𝑓:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
2119, 20nsyl 135 . . . 4 (𝐴𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
2221nexdv 1863 . . 3 (𝐴𝑉 → ¬ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
23 ensym 8002 . . . 4 (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → (𝒫 𝐴 ∩ dom card) ≈ 𝐴)
24 bren 7961 . . . 4 ((𝒫 𝐴 ∩ dom card) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
2523, 24sylib 208 . . 3 (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
2622, 25nsyl 135 . 2 (𝐴𝑉 → ¬ 𝐴 ≈ (𝒫 𝐴 ∩ dom card))
27 brsdom 7975 . 2 (𝐴 ≺ (𝒫 𝐴 ∩ dom card) ↔ (𝐴 ≼ (𝒫 𝐴 ∩ dom card) ∧ ¬ 𝐴 ≈ (𝒫 𝐴 ∩ dom card)))
2814, 26, 27sylanbrc 698 1 (𝐴𝑉𝐴 ≺ (𝒫 𝐴 ∩ dom card))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1482  wex 1703  wcel 1989  wral 2911  Vcvv 3198  cin 3571  wss 3572  𝒫 cpw 4156  {csn 4175   cuni 4434   class class class wbr 4651  {copab 4710   We wwe 5070   × cxp 5110  ccnv 5111  dom cdm 5112  cima 5115  1-1wf1 5883  1-1-ontowf1o 5885  cfv 5886  cen 7949  cdom 7950  csdm 7951  Fincfn 7952  cardccrd 8758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-om 7063  df-1st 7165  df-wrecs 7404  df-recs 7465  df-1o 7557  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-oi 8412  df-card 8762
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator