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Theorem canthnum 9655
Description: The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 8270. 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 4991 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 inex1g 4945 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
3 infpwfidom 9033 . . . 4 ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
41, 2, 33syl 18 . . 3 (𝐴𝑉𝐴 ≼ (𝒫 𝐴 ∩ Fin))
5 inex1g 4945 . . . . 5 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ dom card) ∈ V)
61, 5syl 17 . . . 4 (𝐴𝑉 → (𝒫 𝐴 ∩ dom card) ∈ V)
7 finnum 8956 . . . . . 6 (𝑥 ∈ Fin → 𝑥 ∈ dom card)
87ssriv 3740 . . . . 5 Fin ⊆ dom card
9 sslin 3974 . . . . 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 8159 . . . 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 8166 . . 3 ((𝐴 ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≼ (𝒫 𝐴 ∩ dom card)) → 𝐴 ≼ (𝒫 𝐴 ∩ dom card))
144, 12, 13syl2anc 696 . 2 (𝐴𝑉𝐴 ≼ (𝒫 𝐴 ∩ dom card))
15 eqid 2752 . . . . . . 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 9650 . . . . . 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 2752 . . . . . 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 2752 . . . . . 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 9654 . . . . 5 (𝐴𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
20 f1of1 6289 . . . . 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 2005 . . 3 (𝐴𝑉 → ¬ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
23 ensym 8162 . . . 4 (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → (𝒫 𝐴 ∩ dom card) ≈ 𝐴)
24 bren 8122 . . . 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 8136 . 2 (𝐴 ≺ (𝒫 𝐴 ∩ dom card) ↔ (𝐴 ≼ (𝒫 𝐴 ∩ dom card) ∧ ¬ 𝐴 ≈ (𝒫 𝐴 ∩ dom card)))
2814, 26, 27sylanbrc 701 1 (𝐴𝑉𝐴 ≺ (𝒫 𝐴 ∩ dom card))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1624  wex 1845  wcel 2131  wral 3042  Vcvv 3332  cin 3706  wss 3707  𝒫 cpw 4294  {csn 4313   cuni 4580   class class class wbr 4796  {copab 4856   We wwe 5216   × cxp 5256  ccnv 5257  dom cdm 5258  cima 5261  1-1wf1 6038  1-1-ontowf1o 6040  cfv 6041  cen 8110  cdom 8111  csdm 8112  Fincfn 8113  cardccrd 8943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-om 7223  df-1st 7325  df-wrecs 7568  df-recs 7629  df-1o 7721  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-oi 8572  df-card 8947
This theorem is referenced by: (None)
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