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Theorem tskwe 9363
 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 5244 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 rabexg 5198 . . . 4 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V)
3 incom 4128 . . . . 5 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∩ On) = (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
4 inex1g 5187 . . . . 5 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V → ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∩ On) ∈ V)
53, 4eqeltrrid 2895 . . . 4 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ V)
6 inss1 4155 . . . . . . . . . . 11 (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ On
76sseli 3911 . . . . . . . . . 10 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ On)
8 onelon 6184 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦 ∈ On)
98ancoms 462 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ On) → 𝑦 ∈ On)
107, 9sylan2 595 . . . . . . . . 9 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ On)
11 onelss 6201 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝑦𝑧𝑦𝑧))
1211impcom 411 . . . . . . . . . . . . 13 ((𝑦𝑧𝑧 ∈ On) → 𝑦𝑧)
137, 12sylan2 595 . . . . . . . . . . . 12 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝑧)
14 inss2 4156 . . . . . . . . . . . . . . . . 17 (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}
1514sseli 3911 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
16 breq1 5033 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1716elrab 3628 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ (𝑧 ∈ 𝒫 𝐴𝑧𝐴))
1815, 17sylib 221 . . . . . . . . . . . . . . 15 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → (𝑧 ∈ 𝒫 𝐴𝑧𝐴))
1918simpld 498 . . . . . . . . . . . . . 14 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ 𝒫 𝐴)
2019elpwid 4508 . . . . . . . . . . . . 13 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧𝐴)
2120adantl 485 . . . . . . . . . . . 12 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑧𝐴)
2213, 21sstrd 3925 . . . . . . . . . . 11 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝐴)
23 velpw 4502 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
2422, 23sylibr 237 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ 𝒫 𝐴)
25 vex 3444 . . . . . . . . . . . 12 𝑧 ∈ V
26 ssdomg 8538 . . . . . . . . . . . 12 (𝑧 ∈ V → (𝑦𝑧𝑦𝑧))
2725, 13, 26mpsyl 68 . . . . . . . . . . 11 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝑧)
2818simprd 499 . . . . . . . . . . . 12 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧𝐴)
2928adantl 485 . . . . . . . . . . 11 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑧𝐴)
30 domsdomtr 8636 . . . . . . . . . . 11 ((𝑦𝑧𝑧𝐴) → 𝑦𝐴)
3127, 29, 30syl2anc 587 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝐴)
32 breq1 5033 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
3332elrab 3628 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ (𝑦 ∈ 𝒫 𝐴𝑦𝐴))
3424, 31, 33sylanbrc 586 . . . . . . . . 9 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
3510, 34elind 4121 . . . . . . . 8 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
3635gen2 1798 . . . . . . 7 𝑦𝑧((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
37 dftr2 5138 . . . . . . 7 (Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})))
3836, 37mpbir 234 . . . . . 6 Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
39 ordon 7478 . . . . . 6 Ord On
40 trssord 6176 . . . . . 6 ((Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ On ∧ Ord On) → Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
4138, 6, 39, 40mp3an 1458 . . . . 5 Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
42 elong 6167 . . . . 5 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ V → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On ↔ Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})))
4341, 42mpbiri 261 . . . 4 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ V → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
441, 2, 5, 434syl 19 . . 3 (𝐴𝑉 → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
4544adantr 484 . 2 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
46 simpr 488 . . . . 5 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴)
4714, 46sstrid 3926 . . . 4 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴)
48 ssdomg 8538 . . . . 5 (𝐴𝑉 → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴 → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≼ 𝐴))
4948adantr 484 . . . 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 6177 . . . . 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 1130 . . . . . 6 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
54 elpw2g 5211 . . . . . . . . . 10 (𝐴𝑉 → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴))
5554adantr 484 . . . . . . . . 9 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴))
5647, 55mpbird 260 . . . . . . . 8 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴)
57563adant3 1129 . . . . . . 7 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴)
58 simp3 1135 . . . . . . 7 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴)
59 nfcv 2955 . . . . . . . . 9 𝑥On
60 nfrab1 3337 . . . . . . . . 9 𝑥{𝑥 ∈ 𝒫 𝐴𝑥𝐴}
6159, 60nfin 4143 . . . . . . . 8 𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
62 nfcv 2955 . . . . . . . 8 𝑥𝒫 𝐴
63 nfcv 2955 . . . . . . . . 9 𝑥
64 nfcv 2955 . . . . . . . . 9 𝑥𝐴
6561, 63, 64nfbr 5077 . . . . . . . 8 𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴
66 breq1 5033 . . . . . . . 8 (𝑥 = (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → (𝑥𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
6761, 62, 65, 66elrabf 3624 . . . . . . 7 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
6857, 58, 67sylanbrc 586 . . . . . 6 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
6953, 68elind 4121 . . . . 5 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
70693expia 1118 . . . 4 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴 → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})))
7152, 70mtod 201 . . 3 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴)
72 bren2 8523 . . 3 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴 ↔ ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≼ 𝐴 ∧ ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
7350, 71, 72sylanbrc 586 . 2 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴)
74 isnumi 9359 . 2 (((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴) → 𝐴 ∈ dom card)
7545, 73, 74syl2anc 587 1 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → 𝐴 ∈ dom card)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   ∈ wcel 2111  {crab 3110  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497   class class class wbr 5030  Tr wtr 5136  dom cdm 5519  Ord word 6158  Oncon0 6159   ≈ cen 8489   ≼ cdom 8490   ≺ csdm 8491  cardccrd 9348 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-card 9352 This theorem is referenced by:  tskwe2  10184  grothac  10241
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