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Theorem canthnum 10646
Description: The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9132. 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 5369 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 inex1g 5312 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
3 infpwfidom 10025 . . . 4 ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
41, 2, 33syl 18 . . 3 (𝐴𝑉𝐴 ≼ (𝒫 𝐴 ∩ Fin))
5 inex1g 5312 . . . . 5 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ dom card) ∈ V)
61, 5syl 17 . . . 4 (𝐴𝑉 → (𝒫 𝐴 ∩ dom card) ∈ V)
7 finnum 9945 . . . . . 6 (𝑥 ∈ Fin → 𝑥 ∈ dom card)
87ssriv 3981 . . . . 5 Fin ⊆ dom card
9 sslin 4229 . . . . 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 8998 . . . 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 9005 . . 3 ((𝐴 ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≼ (𝒫 𝐴 ∩ dom card)) → 𝐴 ≼ (𝒫 𝐴 ∩ dom card))
144, 12, 13syl2anc 583 . 2 (𝐴𝑉𝐴 ≼ (𝒫 𝐴 ∩ dom card))
15 eqid 2726 . . . . . . 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 10641 . . . . . 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 2726 . . . . . 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 2726 . . . . . 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 10645 . . . . 5 (𝐴𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
20 f1of1 6826 . . . . 5 (𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴𝑓:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
2119, 20nsyl 140 . . . 4 (𝐴𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
2221nexdv 1931 . . 3 (𝐴𝑉 → ¬ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
23 ensym 9001 . . . 4 (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → (𝒫 𝐴 ∩ dom card) ≈ 𝐴)
24 bren 8951 . . . 4 ((𝒫 𝐴 ∩ dom card) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
2523, 24sylib 217 . . 3 (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
2622, 25nsyl 140 . 2 (𝐴𝑉 → ¬ 𝐴 ≈ (𝒫 𝐴 ∩ dom card))
27 brsdom 8973 . 2 (𝐴 ≺ (𝒫 𝐴 ∩ dom card) ↔ (𝐴 ≼ (𝒫 𝐴 ∩ dom card) ∧ ¬ 𝐴 ≈ (𝒫 𝐴 ∩ dom card)))
2814, 26, 27sylanbrc 582 1 (𝐴𝑉𝐴 ≺ (𝒫 𝐴 ∩ dom card))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1533  wex 1773  wcel 2098  wral 3055  Vcvv 3468  cin 3942  wss 3943  𝒫 cpw 4597  {csn 4623   cuni 4902   class class class wbr 5141  {copab 5203   We wwe 5623   × cxp 5667  ccnv 5668  dom cdm 5669  cima 5672  1-1wf1 6534  1-1-ontowf1o 6536  cfv 6537  cen 8938  cdom 8939  csdm 8940  Fincfn 8941  cardccrd 9932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-card 9936
This theorem is referenced by: (None)
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