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Theorem canthwe 10062
Description: The set of well-orders of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 8654. Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.)
Hypothesis
Ref Expression
canthwe.1 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}
Assertion
Ref Expression
canthwe (𝐴𝑉𝐴𝑂)
Distinct variable groups:   𝑥,𝑟,𝑂   𝑉,𝑟,𝑥   𝐴,𝑟,𝑥

Proof of Theorem canthwe
Dummy variables 𝑢 𝑦 𝑓 𝑣 𝑤 𝑎 𝑠 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1133 . . . . . . . 8 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥𝐴)
2 velpw 4502 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
31, 2sylibr 237 . . . . . . 7 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ 𝒫 𝐴)
4 simp2 1134 . . . . . . . . 9 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ⊆ (𝑥 × 𝑥))
5 xpss12 5534 . . . . . . . . . 10 ((𝑥𝐴𝑥𝐴) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
61, 1, 5syl2anc 587 . . . . . . . . 9 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
74, 6sstrd 3925 . . . . . . . 8 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ⊆ (𝐴 × 𝐴))
8 velpw 4502 . . . . . . . 8 (𝑟 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑟 ⊆ (𝐴 × 𝐴))
97, 8sylibr 237 . . . . . . 7 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
103, 9jca 515 . . . . . 6 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥 ∈ 𝒫 𝐴𝑟 ∈ 𝒫 (𝐴 × 𝐴)))
1110ssopab2i 5402 . . . . 5 {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ⊆ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑟 ∈ 𝒫 (𝐴 × 𝐴))}
12 canthwe.1 . . . . 5 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}
13 df-xp 5525 . . . . 5 (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑟 ∈ 𝒫 (𝐴 × 𝐴))}
1411, 12, 133sstr4i 3958 . . . 4 𝑂 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴))
15 pwexg 5244 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
16 sqxpexg 7457 . . . . . 6 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
1716pwexd 5245 . . . . 5 (𝐴𝑉 → 𝒫 (𝐴 × 𝐴) ∈ V)
1815, 17xpexd 7454 . . . 4 (𝐴𝑉 → (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∈ V)
19 ssexg 5191 . . . 4 ((𝑂 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∧ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∈ V) → 𝑂 ∈ V)
2014, 18, 19sylancr 590 . . 3 (𝐴𝑉𝑂 ∈ V)
21 simpr 488 . . . . . . . 8 ((𝐴𝑉𝑢𝐴) → 𝑢𝐴)
2221snssd 4702 . . . . . . 7 ((𝐴𝑉𝑢𝐴) → {𝑢} ⊆ 𝐴)
23 0ss 4304 . . . . . . . 8 ∅ ⊆ ({𝑢} × {𝑢})
2423a1i 11 . . . . . . 7 ((𝐴𝑉𝑢𝐴) → ∅ ⊆ ({𝑢} × {𝑢}))
25 rel0 5636 . . . . . . . 8 Rel ∅
26 br0 5079 . . . . . . . . 9 ¬ 𝑢𝑢
27 wesn 5604 . . . . . . . . 9 (Rel ∅ → (∅ We {𝑢} ↔ ¬ 𝑢𝑢))
2826, 27mpbiri 261 . . . . . . . 8 (Rel ∅ → ∅ We {𝑢})
2925, 28mp1i 13 . . . . . . 7 ((𝐴𝑉𝑢𝐴) → ∅ We {𝑢})
30 snex 5297 . . . . . . . 8 {𝑢} ∈ V
31 0ex 5175 . . . . . . . 8 ∅ ∈ V
32 simpl 486 . . . . . . . . . 10 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → 𝑥 = {𝑢})
3332sseq1d 3946 . . . . . . . . 9 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑥𝐴 ↔ {𝑢} ⊆ 𝐴))
34 simpr 488 . . . . . . . . . 10 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → 𝑟 = ∅)
3532sqxpeqd 5551 . . . . . . . . . 10 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑥 × 𝑥) = ({𝑢} × {𝑢}))
3634, 35sseq12d 3948 . . . . . . . . 9 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ∅ ⊆ ({𝑢} × {𝑢})))
37 weeq2 5508 . . . . . . . . . 10 (𝑥 = {𝑢} → (𝑟 We 𝑥𝑟 We {𝑢}))
38 weeq1 5507 . . . . . . . . . 10 (𝑟 = ∅ → (𝑟 We {𝑢} ↔ ∅ We {𝑢}))
3937, 38sylan9bb 513 . . . . . . . . 9 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑟 We 𝑥 ↔ ∅ We {𝑢}))
4033, 36, 393anbi123d 1433 . . . . . . . 8 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ ({𝑢} ⊆ 𝐴 ∧ ∅ ⊆ ({𝑢} × {𝑢}) ∧ ∅ We {𝑢})))
4130, 31, 40opelopaba 5388 . . . . . . 7 (⟨{𝑢}, ∅⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ ({𝑢} ⊆ 𝐴 ∧ ∅ ⊆ ({𝑢} × {𝑢}) ∧ ∅ We {𝑢}))
4222, 24, 29, 41syl3anbrc 1340 . . . . . 6 ((𝐴𝑉𝑢𝐴) → ⟨{𝑢}, ∅⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
4342, 12eleqtrrdi 2901 . . . . 5 ((𝐴𝑉𝑢𝐴) → ⟨{𝑢}, ∅⟩ ∈ 𝑂)
4443ex 416 . . . 4 (𝐴𝑉 → (𝑢𝐴 → ⟨{𝑢}, ∅⟩ ∈ 𝑂))
45 eqid 2798 . . . . . . 7 ∅ = ∅
46 snex 5297 . . . . . . . 8 {𝑣} ∈ V
4746, 31opth2 5337 . . . . . . 7 (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ ({𝑢} = {𝑣} ∧ ∅ = ∅))
4845, 47mpbiran2 709 . . . . . 6 (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ {𝑢} = {𝑣})
49 sneqbg 4734 . . . . . . 7 (𝑢 ∈ V → ({𝑢} = {𝑣} ↔ 𝑢 = 𝑣))
5049elv 3446 . . . . . 6 ({𝑢} = {𝑣} ↔ 𝑢 = 𝑣)
5148, 50bitri 278 . . . . 5 (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ 𝑢 = 𝑣)
52512a1i 12 . . . 4 (𝐴𝑉 → ((𝑢𝐴𝑣𝐴) → (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ 𝑢 = 𝑣)))
5344, 52dom2d 8533 . . 3 (𝐴𝑉 → (𝑂 ∈ V → 𝐴𝑂))
5420, 53mpd 15 . 2 (𝐴𝑉𝐴𝑂)
55 eqid 2798 . . . . . . 7 {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))} = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}
5655fpwwe2cbv 10041 . . . . . 6 {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))} = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑤](𝑤𝑓(𝑟 ∩ (𝑤 × 𝑤))) = 𝑦))}
57 eqid 2798 . . . . . 6 dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))} = dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}
58 eqid 2798 . . . . . 6 (({⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}‘ dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}) “ {( dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}𝑓({⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}‘ dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}))}) = (({⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}‘ dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}) “ {( dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}𝑓({⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}‘ dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}))})
5912, 56, 57, 58canthwelem 10061 . . . . 5 (𝐴𝑉 → ¬ 𝑓:𝑂1-1𝐴)
60 f1of1 6589 . . . . 5 (𝑓:𝑂1-1-onto𝐴𝑓:𝑂1-1𝐴)
6159, 60nsyl 142 . . . 4 (𝐴𝑉 → ¬ 𝑓:𝑂1-1-onto𝐴)
6261nexdv 1937 . . 3 (𝐴𝑉 → ¬ ∃𝑓 𝑓:𝑂1-1-onto𝐴)
63 ensym 8541 . . . 4 (𝐴𝑂𝑂𝐴)
64 bren 8501 . . . 4 (𝑂𝐴 ↔ ∃𝑓 𝑓:𝑂1-1-onto𝐴)
6563, 64sylib 221 . . 3 (𝐴𝑂 → ∃𝑓 𝑓:𝑂1-1-onto𝐴)
6662, 65nsyl 142 . 2 (𝐴𝑉 → ¬ 𝐴𝑂)
67 brsdom 8515 . 2 (𝐴𝑂 ↔ (𝐴𝑂 ∧ ¬ 𝐴𝑂))
6854, 66, 67sylanbrc 586 1 (𝐴𝑉𝐴𝑂)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2111  wral 3106  Vcvv 3441  [wsbc 3720  cin 3880  wss 3881  c0 4243  𝒫 cpw 4497  {csn 4525  cop 4531   cuni 4800   class class class wbr 5030  {copab 5092   We wwe 5477   × cxp 5517  ccnv 5518  dom cdm 5519  cima 5522  Rel wrel 5524  1-1wf1 6321  1-1-ontowf1o 6323  cfv 6324  (class class class)co 7135  cen 8489  cdom 8490  csdm 8491
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 7441
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-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-wrecs 7930  df-recs 7991  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-oi 8958
This theorem is referenced by: (None)
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