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Theorem canthnum 10689
Description: The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9170. 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 5378 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 inex1g 5319 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
3 infpwfidom 10068 . . . 4 ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
41, 2, 33syl 18 . . 3 (𝐴𝑉𝐴 ≼ (𝒫 𝐴 ∩ Fin))
5 inex1g 5319 . . . . 5 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ dom card) ∈ V)
61, 5syl 17 . . . 4 (𝐴𝑉 → (𝒫 𝐴 ∩ dom card) ∈ V)
7 finnum 9988 . . . . . 6 (𝑥 ∈ Fin → 𝑥 ∈ dom card)
87ssriv 3987 . . . . 5 Fin ⊆ dom card
9 sslin 4243 . . . . 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 9040 . . . 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 9047 . . 3 ((𝐴 ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≼ (𝒫 𝐴 ∩ dom card)) → 𝐴 ≼ (𝒫 𝐴 ∩ dom card))
144, 12, 13syl2anc 584 . 2 (𝐴𝑉𝐴 ≼ (𝒫 𝐴 ∩ dom card))
15 eqid 2737 . . . . . . 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 10684 . . . . . 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 2737 . . . . . 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 2737 . . . . . 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 10688 . . . . 5 (𝐴𝑉 → ¬ 𝑓:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
20 f1of1 6847 . . . . 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 1936 . . 3 (𝐴𝑉 → ¬ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
23 ensym 9043 . . . 4 (𝐴 ≈ (𝒫 𝐴 ∩ dom card) → (𝒫 𝐴 ∩ dom card) ≈ 𝐴)
24 bren 8995 . . . 4 ((𝒫 𝐴 ∩ dom card) ≈ 𝐴 ↔ ∃𝑓 𝑓:(𝒫 𝐴 ∩ dom card)–1-1-onto𝐴)
2523, 24sylib 218 . . 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 9015 . 2 (𝐴 ≺ (𝒫 𝐴 ∩ dom card) ↔ (𝐴 ≼ (𝒫 𝐴 ∩ dom card) ∧ ¬ 𝐴 ≈ (𝒫 𝐴 ∩ dom card)))
2814, 26, 27sylanbrc 583 1 (𝐴𝑉𝐴 ≺ (𝒫 𝐴 ∩ dom card))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600  {csn 4626   cuni 4907   class class class wbr 5143  {copab 5205   We wwe 5636   × cxp 5683  ccnv 5684  dom cdm 5685  cima 5688  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  cen 8982  cdom 8983  csdm 8984  Fincfn 8985  cardccrd 9975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-card 9979
This theorem is referenced by: (None)
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