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Theorem tskwe 8659
 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 4776 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 rabexg 4739 . . . 4 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V)
3 incom 3767 . . . . 5 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∩ On) = (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
4 inex1g 4729 . . . . 5 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V → ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∩ On) ∈ V)
53, 4syl5eqelr 2693 . . . 4 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ V)
6 inss1 3795 . . . . . . . . . . 11 (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ On
76sseli 3564 . . . . . . . . . 10 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ On)
8 onelon 5665 . . . . . . . . . . 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 5683 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝑦𝑧𝑦𝑧))
1211impcom 445 . . . . . . . . . . . . 13 ((𝑦𝑧𝑧 ∈ On) → 𝑦𝑧)
137, 12sylan2 490 . . . . . . . . . . . 12 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝑧)
14 inss2 3796 . . . . . . . . . . . . . . . . 17 (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}
1514sseli 3564 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
16 breq1 4586 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1716elrab 3331 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ (𝑧 ∈ 𝒫 𝐴𝑧𝐴))
1815, 17sylib 207 . . . . . . . . . . . . . . 15 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → (𝑧 ∈ 𝒫 𝐴𝑧𝐴))
1918simpld 474 . . . . . . . . . . . . . 14 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ 𝒫 𝐴)
2019elpwid 4118 . . . . . . . . . . . . 13 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧𝐴)
2120adantl 481 . . . . . . . . . . . 12 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑧𝐴)
2213, 21sstrd 3578 . . . . . . . . . . 11 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝐴)
23 selpw 4115 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
2422, 23sylibr 223 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ 𝒫 𝐴)
25 vex 3176 . . . . . . . . . . . 12 𝑧 ∈ V
26 ssdomg 7887 . . . . . . . . . . . 12 (𝑧 ∈ V → (𝑦𝑧𝑦𝑧))
2725, 13, 26mpsyl 66 . . . . . . . . . . 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 7980 . . . . . . . . . . 11 ((𝑦𝑧𝑧𝐴) → 𝑦𝐴)
3127, 29, 30syl2anc 691 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝐴)
32 breq1 4586 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
3332elrab 3331 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ (𝑦 ∈ 𝒫 𝐴𝑦𝐴))
3424, 31, 33sylanbrc 695 . . . . . . . . 9 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
3510, 34elind 3760 . . . . . . . 8 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
3635gen2 1714 . . . . . . 7 𝑦𝑧((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
37 dftr2 4682 . . . . . . 7 (Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})))
3836, 37mpbir 220 . . . . . 6 Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
39 ordon 6874 . . . . . 6 Ord On
40 trssord 5657 . . . . . 6 ((Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ On ∧ Ord On) → Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
4138, 6, 39, 40mp3an 1416 . . . . 5 Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
42 elong 5648 . . . . 5 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ V → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On ↔ Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})))
4341, 42mpbiri 247 . . . 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 3579 . . . 4 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴)
48 ssdomg 7887 . . . . 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 5658 . . . . 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 1075 . . . . . 6 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
54 elpw2g 4754 . . . . . . . . . 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 246 . . . . . . . 8 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴)
57563adant3 1074 . . . . . . 7 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴)
58 simp3 1056 . . . . . . 7 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴)
59 nfcv 2751 . . . . . . . . 9 𝑥On
60 nfrab1 3099 . . . . . . . . 9 𝑥{𝑥 ∈ 𝒫 𝐴𝑥𝐴}
6159, 60nfin 3782 . . . . . . . 8 𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
62 nfcv 2751 . . . . . . . 8 𝑥𝒫 𝐴
63 nfcv 2751 . . . . . . . . 9 𝑥
64 nfcv 2751 . . . . . . . . 9 𝑥𝐴
6561, 63, 64nfbr 4629 . . . . . . . 8 𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴
66 breq1 4586 . . . . . . . 8 (𝑥 = (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → (𝑥𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
6761, 62, 65, 66elrabf 3329 . . . . . . 7 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
6857, 58, 67sylanbrc 695 . . . . . 6 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
6953, 68elind 3760 . . . . 5 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
70693expia 1259 . . . 4 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴 → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})))
7152, 70mtod 188 . . 3 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴)
72 bren2 7872 . . 3 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴 ↔ ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≼ 𝐴 ∧ ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
7350, 71, 72sylanbrc 695 . 2 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴)
74 isnumi 8655 . 2 (((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴) → 𝐴 ∈ dom card)
7545, 73, 74syl2anc 691 1 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → 𝐴 ∈ dom card)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583  Tr wtr 4680  dom cdm 5038  Ord word 5639  Oncon0 5640   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  cardccrd 8644 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-card 8648 This theorem is referenced by:  tskwe2  9474  grothac  9531
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