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Theorem canthwe 10061
Description: The set of well-orders of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 8658. 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 1128 . . . . . . . 8 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥𝐴)
2 velpw 4543 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
31, 2sylibr 235 . . . . . . 7 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ 𝒫 𝐴)
4 simp2 1129 . . . . . . . . 9 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ⊆ (𝑥 × 𝑥))
5 xpss12 5563 . . . . . . . . . 10 ((𝑥𝐴𝑥𝐴) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
61, 1, 5syl2anc 584 . . . . . . . . 9 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
74, 6sstrd 3974 . . . . . . . 8 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ⊆ (𝐴 × 𝐴))
8 velpw 4543 . . . . . . . 8 (𝑟 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑟 ⊆ (𝐴 × 𝐴))
97, 8sylibr 235 . . . . . . 7 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
103, 9jca 512 . . . . . 6 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥 ∈ 𝒫 𝐴𝑟 ∈ 𝒫 (𝐴 × 𝐴)))
1110ssopab2i 5428 . . . . 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 5554 . . . . 5 (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑟 ∈ 𝒫 (𝐴 × 𝐴))}
1411, 12, 133sstr4i 4007 . . . 4 𝑂 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴))
15 pwexg 5270 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
16 sqxpexg 7466 . . . . . 6 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
1716pwexd 5271 . . . . 5 (𝐴𝑉 → 𝒫 (𝐴 × 𝐴) ∈ V)
1815, 17xpexd 7463 . . . 4 (𝐴𝑉 → (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∈ V)
19 ssexg 5218 . . . 4 ((𝑂 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∧ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∈ V) → 𝑂 ∈ V)
2014, 18, 19sylancr 587 . . 3 (𝐴𝑉𝑂 ∈ V)
21 simpr 485 . . . . . . . 8 ((𝐴𝑉𝑢𝐴) → 𝑢𝐴)
2221snssd 4734 . . . . . . 7 ((𝐴𝑉𝑢𝐴) → {𝑢} ⊆ 𝐴)
23 0ss 4347 . . . . . . . 8 ∅ ⊆ ({𝑢} × {𝑢})
2423a1i 11 . . . . . . 7 ((𝐴𝑉𝑢𝐴) → ∅ ⊆ ({𝑢} × {𝑢}))
25 rel0 5665 . . . . . . . 8 Rel ∅
26 br0 5106 . . . . . . . . 9 ¬ 𝑢𝑢
27 wesn 5633 . . . . . . . . 9 (Rel ∅ → (∅ We {𝑢} ↔ ¬ 𝑢𝑢))
2826, 27mpbiri 259 . . . . . . . 8 (Rel ∅ → ∅ We {𝑢})
2925, 28mp1i 13 . . . . . . 7 ((𝐴𝑉𝑢𝐴) → ∅ We {𝑢})
30 snex 5322 . . . . . . . 8 {𝑢} ∈ V
31 0ex 5202 . . . . . . . 8 ∅ ∈ V
32 simpl 483 . . . . . . . . . 10 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → 𝑥 = {𝑢})
3332sseq1d 3995 . . . . . . . . 9 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑥𝐴 ↔ {𝑢} ⊆ 𝐴))
34 simpr 485 . . . . . . . . . 10 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → 𝑟 = ∅)
3532sqxpeqd 5580 . . . . . . . . . 10 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑥 × 𝑥) = ({𝑢} × {𝑢}))
3634, 35sseq12d 3997 . . . . . . . . 9 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ∅ ⊆ ({𝑢} × {𝑢})))
37 weeq2 5537 . . . . . . . . . 10 (𝑥 = {𝑢} → (𝑟 We 𝑥𝑟 We {𝑢}))
38 weeq1 5536 . . . . . . . . . 10 (𝑟 = ∅ → (𝑟 We {𝑢} ↔ ∅ We {𝑢}))
3937, 38sylan9bb 510 . . . . . . . . 9 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑟 We 𝑥 ↔ ∅ We {𝑢}))
4033, 36, 393anbi123d 1427 . . . . . . . 8 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ ({𝑢} ⊆ 𝐴 ∧ ∅ ⊆ ({𝑢} × {𝑢}) ∧ ∅ We {𝑢})))
4130, 31, 40opelopaba 5414 . . . . . . 7 (⟨{𝑢}, ∅⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ ({𝑢} ⊆ 𝐴 ∧ ∅ ⊆ ({𝑢} × {𝑢}) ∧ ∅ We {𝑢}))
4222, 24, 29, 41syl3anbrc 1335 . . . . . 6 ((𝐴𝑉𝑢𝐴) → ⟨{𝑢}, ∅⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
4342, 12eleqtrrdi 2921 . . . . 5 ((𝐴𝑉𝑢𝐴) → ⟨{𝑢}, ∅⟩ ∈ 𝑂)
4443ex 413 . . . 4 (𝐴𝑉 → (𝑢𝐴 → ⟨{𝑢}, ∅⟩ ∈ 𝑂))
45 eqid 2818 . . . . . . 7 ∅ = ∅
46 snex 5322 . . . . . . . 8 {𝑣} ∈ V
4746, 31opth2 5363 . . . . . . 7 (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ ({𝑢} = {𝑣} ∧ ∅ = ∅))
4845, 47mpbiran2 706 . . . . . 6 (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ {𝑢} = {𝑣})
49 sneqbg 4766 . . . . . . 7 (𝑢 ∈ V → ({𝑢} = {𝑣} ↔ 𝑢 = 𝑣))
5049elv 3497 . . . . . 6 ({𝑢} = {𝑣} ↔ 𝑢 = 𝑣)
5148, 50bitri 276 . . . . 5 (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ 𝑢 = 𝑣)
52512a1i 12 . . . 4 (𝐴𝑉 → ((𝑢𝐴𝑣𝐴) → (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ 𝑢 = 𝑣)))
5344, 52dom2d 8538 . . 3 (𝐴𝑉 → (𝑂 ∈ V → 𝐴𝑂))
5420, 53mpd 15 . 2 (𝐴𝑉𝐴𝑂)
55 eqid 2818 . . . . . . 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 10040 . . . . . 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 2818 . . . . . 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 2818 . . . . . 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 10060 . . . . 5 (𝐴𝑉 → ¬ 𝑓:𝑂1-1𝐴)
60 f1of1 6607 . . . . 5 (𝑓:𝑂1-1-onto𝐴𝑓:𝑂1-1𝐴)
6159, 60nsyl 142 . . . 4 (𝐴𝑉 → ¬ 𝑓:𝑂1-1-onto𝐴)
6261nexdv 1928 . . 3 (𝐴𝑉 → ¬ ∃𝑓 𝑓:𝑂1-1-onto𝐴)
63 ensym 8546 . . . 4 (𝐴𝑂𝑂𝐴)
64 bren 8506 . . . 4 (𝑂𝐴 ↔ ∃𝑓 𝑓:𝑂1-1-onto𝐴)
6563, 64sylib 219 . . 3 (𝐴𝑂 → ∃𝑓 𝑓:𝑂1-1-onto𝐴)
6662, 65nsyl 142 . 2 (𝐴𝑉 → ¬ 𝐴𝑂)
67 brsdom 8520 . 2 (𝐴𝑂 ↔ (𝐴𝑂 ∧ ¬ 𝐴𝑂))
6854, 66, 67sylanbrc 583 1 (𝐴𝑉𝐴𝑂)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wral 3135  Vcvv 3492  [wsbc 3769  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535  {csn 4557  cop 4563   cuni 4830   class class class wbr 5057  {copab 5119   We wwe 5506   × cxp 5546  ccnv 5547  dom cdm 5548  cima 5551  Rel wrel 5553  1-1wf1 6345  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  cen 8494  cdom 8495  csdm 8496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-wrecs 7936  df-recs 7997  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-oi 8962
This theorem is referenced by: (None)
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