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Theorem tskwe 8814
Description: A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
tskwe ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → 𝐴 ∈ dom card)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem tskwe
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4880 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 rabexg 4844 . . . 4 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V)
3 incom 3838 . . . . 5 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∩ On) = (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
4 inex1g 4834 . . . . 5 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V → ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∩ On) ∈ V)
53, 4syl5eqelr 2735 . . . 4 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ V)
6 inss1 3866 . . . . . . . . . . 11 (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ On
76sseli 3632 . . . . . . . . . 10 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ On)
8 onelon 5786 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦 ∈ On)
98ancoms 468 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ On) → 𝑦 ∈ On)
107, 9sylan2 490 . . . . . . . . 9 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ On)
11 onelss 5804 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝑦𝑧𝑦𝑧))
1211impcom 445 . . . . . . . . . . . . 13 ((𝑦𝑧𝑧 ∈ On) → 𝑦𝑧)
137, 12sylan2 490 . . . . . . . . . . . 12 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝑧)
14 inss2 3867 . . . . . . . . . . . . . . . . 17 (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}
1514sseli 3632 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
16 breq1 4688 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1716elrab 3396 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ (𝑧 ∈ 𝒫 𝐴𝑧𝐴))
1815, 17sylib 208 . . . . . . . . . . . . . . 15 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → (𝑧 ∈ 𝒫 𝐴𝑧𝐴))
1918simpld 474 . . . . . . . . . . . . . 14 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ 𝒫 𝐴)
2019elpwid 4203 . . . . . . . . . . . . 13 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧𝐴)
2120adantl 481 . . . . . . . . . . . 12 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑧𝐴)
2213, 21sstrd 3646 . . . . . . . . . . 11 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝐴)
23 selpw 4198 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
2422, 23sylibr 224 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ 𝒫 𝐴)
25 vex 3234 . . . . . . . . . . . 12 𝑧 ∈ V
26 ssdomg 8043 . . . . . . . . . . . 12 (𝑧 ∈ V → (𝑦𝑧𝑦𝑧))
2725, 13, 26mpsyl 68 . . . . . . . . . . 11 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝑧)
2818simprd 478 . . . . . . . . . . . 12 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧𝐴)
2928adantl 481 . . . . . . . . . . 11 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑧𝐴)
30 domsdomtr 8136 . . . . . . . . . . 11 ((𝑦𝑧𝑧𝐴) → 𝑦𝐴)
3127, 29, 30syl2anc 694 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝐴)
32 breq1 4688 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
3332elrab 3396 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ (𝑦 ∈ 𝒫 𝐴𝑦𝐴))
3424, 31, 33sylanbrc 699 . . . . . . . . 9 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
3510, 34elind 3831 . . . . . . . 8 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
3635gen2 1763 . . . . . . 7 𝑦𝑧((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
37 dftr2 4787 . . . . . . 7 (Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})))
3836, 37mpbir 221 . . . . . 6 Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
39 ordon 7024 . . . . . 6 Ord On
40 trssord 5778 . . . . . 6 ((Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ On ∧ Ord On) → Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
4138, 6, 39, 40mp3an 1464 . . . . 5 Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
42 elong 5769 . . . . 5 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ V → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On ↔ Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})))
4341, 42mpbiri 248 . . . 4 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ V → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
441, 2, 5, 434syl 19 . . 3 (𝐴𝑉 → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
4544adantr 480 . 2 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
46 simpr 476 . . . . 5 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴)
4714, 46syl5ss 3647 . . . 4 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴)
48 ssdomg 8043 . . . . 5 (𝐴𝑉 → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴 → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≼ 𝐴))
4948adantr 480 . . . 4 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴 → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≼ 𝐴))
5047, 49mpd 15 . . 3 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≼ 𝐴)
51 ordirr 5779 . . . . 5 (Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
5241, 51mp1i 13 . . . 4 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
53443ad2ant1 1102 . . . . . 6 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
54 elpw2g 4857 . . . . . . . . . 10 (𝐴𝑉 → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴))
5554adantr 480 . . . . . . . . 9 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴))
5647, 55mpbird 247 . . . . . . . 8 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴)
57563adant3 1101 . . . . . . 7 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴)
58 simp3 1083 . . . . . . 7 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴)
59 nfcv 2793 . . . . . . . . 9 𝑥On
60 nfrab1 3152 . . . . . . . . 9 𝑥{𝑥 ∈ 𝒫 𝐴𝑥𝐴}
6159, 60nfin 3853 . . . . . . . 8 𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
62 nfcv 2793 . . . . . . . 8 𝑥𝒫 𝐴
63 nfcv 2793 . . . . . . . . 9 𝑥
64 nfcv 2793 . . . . . . . . 9 𝑥𝐴
6561, 63, 64nfbr 4732 . . . . . . . 8 𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴
66 breq1 4688 . . . . . . . 8 (𝑥 = (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → (𝑥𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
6761, 62, 65, 66elrabf 3392 . . . . . . 7 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
6857, 58, 67sylanbrc 699 . . . . . 6 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
6953, 68elind 3831 . . . . 5 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
70693expia 1286 . . . 4 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴 → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})))
7152, 70mtod 189 . . 3 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴)
72 bren2 8028 . . 3 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴 ↔ ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≼ 𝐴 ∧ ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
7350, 71, 72sylanbrc 699 . 2 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴)
74 isnumi 8810 . 2 (((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴) → 𝐴 ∈ dom card)
7545, 73, 74syl2anc 694 1 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → 𝐴 ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054  wal 1521  wcel 2030  {crab 2945  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191   class class class wbr 4685  Tr wtr 4785  dom cdm 5143  Ord word 5760  Oncon0 5761  cen 7994  cdom 7995  csdm 7996  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-card 8803
This theorem is referenced by:  tskwe2  9633  grothac  9690
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