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Theorem canthwe 9470
Description: The set of well-orders of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 8110. 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 1060 . . . . . . . 8 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥𝐴)
2 selpw 4163 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
31, 2sylibr 224 . . . . . . 7 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ 𝒫 𝐴)
4 simp2 1061 . . . . . . . . 9 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ⊆ (𝑥 × 𝑥))
5 xpss12 5223 . . . . . . . . . 10 ((𝑥𝐴𝑥𝐴) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
61, 1, 5syl2anc 693 . . . . . . . . 9 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
74, 6sstrd 3611 . . . . . . . 8 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ⊆ (𝐴 × 𝐴))
8 selpw 4163 . . . . . . . 8 (𝑟 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑟 ⊆ (𝐴 × 𝐴))
97, 8sylibr 224 . . . . . . 7 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
103, 9jca 554 . . . . . 6 ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥 ∈ 𝒫 𝐴𝑟 ∈ 𝒫 (𝐴 × 𝐴)))
1110ssopab2i 5001 . . . . 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 5118 . . . . 5 (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ∈ 𝒫 𝐴𝑟 ∈ 𝒫 (𝐴 × 𝐴))}
1411, 12, 133sstr4i 3642 . . . 4 𝑂 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴))
15 pwexg 4848 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
16 sqxpexg 6960 . . . . . 6 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
17 pwexg 4848 . . . . . 6 ((𝐴 × 𝐴) ∈ V → 𝒫 (𝐴 × 𝐴) ∈ V)
1816, 17syl 17 . . . . 5 (𝐴𝑉 → 𝒫 (𝐴 × 𝐴) ∈ V)
19 xpexg 6957 . . . . 5 ((𝒫 𝐴 ∈ V ∧ 𝒫 (𝐴 × 𝐴) ∈ V) → (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∈ V)
2015, 18, 19syl2anc 693 . . . 4 (𝐴𝑉 → (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∈ V)
21 ssexg 4802 . . . 4 ((𝑂 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∧ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∈ V) → 𝑂 ∈ V)
2214, 20, 21sylancr 695 . . 3 (𝐴𝑉𝑂 ∈ V)
23 simpr 477 . . . . . . . 8 ((𝐴𝑉𝑢𝐴) → 𝑢𝐴)
2423snssd 4338 . . . . . . 7 ((𝐴𝑉𝑢𝐴) → {𝑢} ⊆ 𝐴)
25 0ss 3970 . . . . . . . 8 ∅ ⊆ ({𝑢} × {𝑢})
2625a1i 11 . . . . . . 7 ((𝐴𝑉𝑢𝐴) → ∅ ⊆ ({𝑢} × {𝑢}))
27 rel0 5241 . . . . . . . 8 Rel ∅
28 br0 4699 . . . . . . . . 9 ¬ 𝑢𝑢
29 wesn 5188 . . . . . . . . 9 (Rel ∅ → (∅ We {𝑢} ↔ ¬ 𝑢𝑢))
3028, 29mpbiri 248 . . . . . . . 8 (Rel ∅ → ∅ We {𝑢})
3127, 30mp1i 13 . . . . . . 7 ((𝐴𝑉𝑢𝐴) → ∅ We {𝑢})
32 snex 4906 . . . . . . . 8 {𝑢} ∈ V
33 0ex 4788 . . . . . . . 8 ∅ ∈ V
34 simpl 473 . . . . . . . . . 10 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → 𝑥 = {𝑢})
3534sseq1d 3630 . . . . . . . . 9 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑥𝐴 ↔ {𝑢} ⊆ 𝐴))
36 simpr 477 . . . . . . . . . 10 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → 𝑟 = ∅)
3734sqxpeqd 5139 . . . . . . . . . 10 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑥 × 𝑥) = ({𝑢} × {𝑢}))
3836, 37sseq12d 3632 . . . . . . . . 9 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ∅ ⊆ ({𝑢} × {𝑢})))
39 weeq2 5101 . . . . . . . . . 10 (𝑥 = {𝑢} → (𝑟 We 𝑥𝑟 We {𝑢}))
40 weeq1 5100 . . . . . . . . . 10 (𝑟 = ∅ → (𝑟 We {𝑢} ↔ ∅ We {𝑢}))
4139, 40sylan9bb 736 . . . . . . . . 9 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑟 We 𝑥 ↔ ∅ We {𝑢}))
4235, 38, 413anbi123d 1398 . . . . . . . 8 ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ ({𝑢} ⊆ 𝐴 ∧ ∅ ⊆ ({𝑢} × {𝑢}) ∧ ∅ We {𝑢})))
4332, 33, 42opelopaba 4989 . . . . . . 7 (⟨{𝑢}, ∅⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ ({𝑢} ⊆ 𝐴 ∧ ∅ ⊆ ({𝑢} × {𝑢}) ∧ ∅ We {𝑢}))
4424, 26, 31, 43syl3anbrc 1245 . . . . . 6 ((𝐴𝑉𝑢𝐴) → ⟨{𝑢}, ∅⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
4544, 12syl6eleqr 2711 . . . . 5 ((𝐴𝑉𝑢𝐴) → ⟨{𝑢}, ∅⟩ ∈ 𝑂)
4645ex 450 . . . 4 (𝐴𝑉 → (𝑢𝐴 → ⟨{𝑢}, ∅⟩ ∈ 𝑂))
47 eqid 2621 . . . . . . 7 ∅ = ∅
48 snex 4906 . . . . . . . 8 {𝑣} ∈ V
4948, 33opth2 4947 . . . . . . 7 (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ ({𝑢} = {𝑣} ∧ ∅ = ∅))
5047, 49mpbiran2 954 . . . . . 6 (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ {𝑢} = {𝑣})
51 vex 3201 . . . . . . 7 𝑢 ∈ V
52 sneqbg 4372 . . . . . . 7 (𝑢 ∈ V → ({𝑢} = {𝑣} ↔ 𝑢 = 𝑣))
5351, 52ax-mp 5 . . . . . 6 ({𝑢} = {𝑣} ↔ 𝑢 = 𝑣)
5450, 53bitri 264 . . . . 5 (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ 𝑢 = 𝑣)
55542a1i 12 . . . 4 (𝐴𝑉 → ((𝑢𝐴𝑣𝐴) → (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ 𝑢 = 𝑣)))
5646, 55dom2d 7993 . . 3 (𝐴𝑉 → (𝑂 ∈ V → 𝐴𝑂))
5722, 56mpd 15 . 2 (𝐴𝑉𝐴𝑂)
58 eqid 2621 . . . . . . 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))}
5958fpwwe2cbv 9449 . . . . . 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))}
60 eqid 2621 . . . . . 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))}
61 eqid 2621 . . . . . 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))}))})
6212, 59, 60, 61canthwelem 9469 . . . . 5 (𝐴𝑉 → ¬ 𝑓:𝑂1-1𝐴)
63 f1of1 6134 . . . . 5 (𝑓:𝑂1-1-onto𝐴𝑓:𝑂1-1𝐴)
6462, 63nsyl 135 . . . 4 (𝐴𝑉 → ¬ 𝑓:𝑂1-1-onto𝐴)
6564nexdv 1863 . . 3 (𝐴𝑉 → ¬ ∃𝑓 𝑓:𝑂1-1-onto𝐴)
66 ensym 8002 . . . 4 (𝐴𝑂𝑂𝐴)
67 bren 7961 . . . 4 (𝑂𝐴 ↔ ∃𝑓 𝑓:𝑂1-1-onto𝐴)
6866, 67sylib 208 . . 3 (𝐴𝑂 → ∃𝑓 𝑓:𝑂1-1-onto𝐴)
6965, 68nsyl 135 . 2 (𝐴𝑉 → ¬ 𝐴𝑂)
70 brsdom 7975 . 2 (𝐴𝑂 ↔ (𝐴𝑂 ∧ ¬ 𝐴𝑂))
7157, 69, 70sylanbrc 698 1 (𝐴𝑉𝐴𝑂)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1037   = wceq 1482  wex 1703  wcel 1989  wral 2911  Vcvv 3198  [wsbc 3433  cin 3571  wss 3572  c0 3913  𝒫 cpw 4156  {csn 4175  cop 4181   cuni 4434   class class class wbr 4651  {copab 4710   We wwe 5070   × cxp 5110  ccnv 5111  dom cdm 5112  cima 5115  Rel wrel 5117  1-1wf1 5883  1-1-ontowf1o 5885  cfv 5886  (class class class)co 6647  cen 7949  cdom 7950  csdm 7951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-wrecs 7404  df-recs 7465  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-oi 8412
This theorem is referenced by: (None)
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