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Theorem tskwe 9027
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 5014 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 rabexg 4972 . . . 4 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V)
3 incom 3967 . . . . 5 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∩ On) = (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
4 inex1g 4962 . . . . 5 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V → ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∩ On) ∈ V)
53, 4syl5eqelr 2849 . . . 4 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ V)
6 inss1 3992 . . . . . . . . . . 11 (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ On
76sseli 3757 . . . . . . . . . 10 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ On)
8 onelon 5933 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦 ∈ On)
98ancoms 450 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ On) → 𝑦 ∈ On)
107, 9sylan2 586 . . . . . . . . 9 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ On)
11 onelss 5950 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝑦𝑧𝑦𝑧))
1211impcom 396 . . . . . . . . . . . . 13 ((𝑦𝑧𝑧 ∈ On) → 𝑦𝑧)
137, 12sylan2 586 . . . . . . . . . . . 12 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝑧)
14 inss2 3993 . . . . . . . . . . . . . . . . 17 (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}
1514sseli 3757 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
16 breq1 4812 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1716elrab 3519 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ (𝑧 ∈ 𝒫 𝐴𝑧𝐴))
1815, 17sylib 209 . . . . . . . . . . . . . . 15 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → (𝑧 ∈ 𝒫 𝐴𝑧𝐴))
1918simpld 488 . . . . . . . . . . . . . 14 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ 𝒫 𝐴)
2019elpwid 4327 . . . . . . . . . . . . 13 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧𝐴)
2120adantl 473 . . . . . . . . . . . 12 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑧𝐴)
2213, 21sstrd 3771 . . . . . . . . . . 11 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝐴)
23 selpw 4322 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
2422, 23sylibr 225 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ 𝒫 𝐴)
25 vex 3353 . . . . . . . . . . . 12 𝑧 ∈ V
26 ssdomg 8206 . . . . . . . . . . . 12 (𝑧 ∈ V → (𝑦𝑧𝑦𝑧))
2725, 13, 26mpsyl 68 . . . . . . . . . . 11 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝑧)
2818simprd 489 . . . . . . . . . . . 12 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧𝐴)
2928adantl 473 . . . . . . . . . . 11 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑧𝐴)
30 domsdomtr 8302 . . . . . . . . . . 11 ((𝑦𝑧𝑧𝐴) → 𝑦𝐴)
3127, 29, 30syl2anc 579 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝐴)
32 breq1 4812 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
3332elrab 3519 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ (𝑦 ∈ 𝒫 𝐴𝑦𝐴))
3424, 31, 33sylanbrc 578 . . . . . . . . 9 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
3510, 34elind 3960 . . . . . . . 8 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
3635gen2 1891 . . . . . . 7 𝑦𝑧((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
37 dftr2 4913 . . . . . . 7 (Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})))
3836, 37mpbir 222 . . . . . 6 Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
39 ordon 7180 . . . . . 6 Ord On
40 trssord 5925 . . . . . 6 ((Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ On ∧ Ord On) → Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
4138, 6, 39, 40mp3an 1585 . . . . 5 Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
42 elong 5916 . . . . 5 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ V → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On ↔ Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})))
4341, 42mpbiri 249 . . . 4 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ V → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
441, 2, 5, 434syl 19 . . 3 (𝐴𝑉 → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
4544adantr 472 . 2 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
46 simpr 477 . . . . 5 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴)
4714, 46syl5ss 3772 . . . 4 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴)
48 ssdomg 8206 . . . . 5 (𝐴𝑉 → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴 → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≼ 𝐴))
4948adantr 472 . . . 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 5926 . . . . 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 1163 . . . . . 6 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
54 elpw2g 4985 . . . . . . . . . 10 (𝐴𝑉 → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴))
5554adantr 472 . . . . . . . . 9 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴))
5647, 55mpbird 248 . . . . . . . 8 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴)
57563adant3 1162 . . . . . . 7 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴)
58 simp3 1168 . . . . . . 7 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴)
59 nfcv 2907 . . . . . . . . 9 𝑥On
60 nfrab1 3270 . . . . . . . . 9 𝑥{𝑥 ∈ 𝒫 𝐴𝑥𝐴}
6159, 60nfin 3980 . . . . . . . 8 𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
62 nfcv 2907 . . . . . . . 8 𝑥𝒫 𝐴
63 nfcv 2907 . . . . . . . . 9 𝑥
64 nfcv 2907 . . . . . . . . 9 𝑥𝐴
6561, 63, 64nfbr 4856 . . . . . . . 8 𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴
66 breq1 4812 . . . . . . . 8 (𝑥 = (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → (𝑥𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
6761, 62, 65, 66elrabf 3515 . . . . . . 7 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
6857, 58, 67sylanbrc 578 . . . . . 6 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
6953, 68elind 3960 . . . . 5 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
70693expia 1150 . . . 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 8191 . . 3 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴 ↔ ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≼ 𝐴 ∧ ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
7350, 71, 72sylanbrc 578 . 2 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴)
74 isnumi 9023 . 2 (((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴) → 𝐴 ∈ dom card)
7545, 73, 74syl2anc 579 1 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → 𝐴 ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107  wal 1650  wcel 2155  {crab 3059  Vcvv 3350  cin 3731  wss 3732  𝒫 cpw 4315   class class class wbr 4809  Tr wtr 4911  dom cdm 5277  Ord word 5907  Oncon0 5908  cen 8157  cdom 8158  csdm 8159  cardccrd 9012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-card 9016
This theorem is referenced by:  tskwe2  9848  grothac  9905
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