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Theorem canthnum 10062
 Description: The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 8656. 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 5244 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 inex1g 5187 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
3 infpwfidom 9441 . . . 4 ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
41, 2, 33syl 18 . . 3 (𝐴𝑉𝐴 ≼ (𝒫 𝐴 ∩ Fin))
5 inex1g 5187 . . . . 5 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ dom card) ∈ V)
61, 5syl 17 . . . 4 (𝐴𝑉 → (𝒫 𝐴 ∩ dom card) ∈ V)
7 finnum 9363 . . . . . 6 (𝑥 ∈ Fin → 𝑥 ∈ dom card)
87ssriv 3919 . . . . 5 Fin ⊆ dom card
9 sslin 4161 . . . . 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 8540 . . . 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 8547 . . 3 ((𝐴 ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≼ (𝒫 𝐴 ∩ dom card)) → 𝐴 ≼ (𝒫 𝐴 ∩ dom card))
144, 12, 13syl2anc 587 . 2 (𝐴𝑉𝐴 ≼ (𝒫 𝐴 ∩ dom card))
15 eqid 2798 . . . . . . 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 10057 . . . . . 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 2798 . . . . . 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 2798 . . . . . 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 10061 . . . . 5 (𝐴𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
20 f1of1 6589 . . . . 5 (𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴𝑓:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
2119, 20nsyl 142 . . . 4 (𝐴𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
2221nexdv 1937 . . 3 (𝐴𝑉 → ¬ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
23 ensym 8543 . . . 4 (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → (𝒫 𝐴 ∩ dom card) ≈ 𝐴)
24 bren 8503 . . . 4 ((𝒫 𝐴 ∩ dom card) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
2523, 24sylib 221 . . 3 (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
2622, 25nsyl 142 . 2 (𝐴𝑉 → ¬ 𝐴 ≈ (𝒫 𝐴 ∩ dom card))
27 brsdom 8517 . 2 (𝐴 ≺ (𝒫 𝐴 ∩ dom card) ↔ (𝐴 ≼ (𝒫 𝐴 ∩ dom card) ∧ ¬ 𝐴 ≈ (𝒫 𝐴 ∩ dom card)))
2814, 26, 27sylanbrc 586 1 (𝐴𝑉𝐴 ≺ (𝒫 𝐴 ∩ dom card))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  {csn 4525  ∪ cuni 4800   class class class wbr 5030  {copab 5092   We wwe 5477   × cxp 5517  ◡ccnv 5518  dom cdm 5519   “ cima 5522  –1-1→wf1 6321  –1-1-onto→wf1o 6323  ‘cfv 6324   ≈ cen 8491   ≼ cdom 8492   ≺ csdm 8493  Fincfn 8494  cardccrd 9350 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-om 7563  df-1st 7673  df-wrecs 7932  df-recs 7993  df-1o 8087  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-oi 8960  df-card 9354 This theorem is referenced by: (None)
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