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Theorem tskwe 9225
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 5170 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 rabexg 5125 . . . 4 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V)
3 incom 4099 . . . . 5 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∩ On) = (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
4 inex1g 5114 . . . . 5 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V → ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∩ On) ∈ V)
53, 4syl5eqelr 2888 . . . 4 ({𝑥 ∈ 𝒫 𝐴𝑥𝐴} ∈ V → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ V)
6 inss1 4125 . . . . . . . . . . 11 (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ On
76sseli 3885 . . . . . . . . . 10 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ On)
8 onelon 6091 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑦𝑧) → 𝑦 ∈ On)
98ancoms 459 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ On) → 𝑦 ∈ On)
107, 9sylan2 592 . . . . . . . . 9 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ On)
11 onelss 6108 . . . . . . . . . . . . . 14 (𝑧 ∈ On → (𝑦𝑧𝑦𝑧))
1211impcom 408 . . . . . . . . . . . . 13 ((𝑦𝑧𝑧 ∈ On) → 𝑦𝑧)
137, 12sylan2 592 . . . . . . . . . . . 12 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝑧)
14 inss2 4126 . . . . . . . . . . . . . . . . 17 (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}
1514sseli 3885 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
16 breq1 4965 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1716elrab 3618 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ (𝑧 ∈ 𝒫 𝐴𝑧𝐴))
1815, 17sylib 219 . . . . . . . . . . . . . . 15 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → (𝑧 ∈ 𝒫 𝐴𝑧𝐴))
1918simpld 495 . . . . . . . . . . . . . 14 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧 ∈ 𝒫 𝐴)
2019elpwid 4465 . . . . . . . . . . . . 13 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧𝐴)
2120adantl 482 . . . . . . . . . . . 12 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑧𝐴)
2213, 21sstrd 3899 . . . . . . . . . . 11 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝐴)
23 selpw 4460 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
2422, 23sylibr 235 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ 𝒫 𝐴)
25 vex 3440 . . . . . . . . . . . 12 𝑧 ∈ V
26 ssdomg 8403 . . . . . . . . . . . 12 (𝑧 ∈ V → (𝑦𝑧𝑦𝑧))
2725, 13, 26mpsyl 68 . . . . . . . . . . 11 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝑧)
2818simprd 496 . . . . . . . . . . . 12 (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → 𝑧𝐴)
2928adantl 482 . . . . . . . . . . 11 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑧𝐴)
30 domsdomtr 8499 . . . . . . . . . . 11 ((𝑦𝑧𝑧𝐴) → 𝑦𝐴)
3127, 29, 30syl2anc 584 . . . . . . . . . 10 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦𝐴)
32 breq1 4965 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
3332elrab 3618 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ (𝑦 ∈ 𝒫 𝐴𝑦𝐴))
3424, 31, 33sylanbrc 583 . . . . . . . . 9 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
3510, 34elind 4092 . . . . . . . 8 ((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
3635gen2 1778 . . . . . . 7 𝑦𝑧((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
37 dftr2 5065 . . . . . . 7 (Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ↔ ∀𝑦𝑧((𝑦𝑧𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})))
3836, 37mpbir 232 . . . . . 6 Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
39 ordon 7354 . . . . . 6 Ord On
40 trssord 6083 . . . . . 6 ((Tr (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ On ∧ Ord On) → Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
4138, 6, 39, 40mp3an 1453 . . . . 5 Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
42 elong 6074 . . . . 5 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ V → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On ↔ Ord (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})))
4341, 42mpbiri 259 . . . 4 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ V → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
441, 2, 5, 434syl 19 . . 3 (𝐴𝑉 → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
4544adantr 481 . 2 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
46 simpr 485 . . . . 5 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴)
4714, 46syl5ss 3900 . . . 4 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴)
48 ssdomg 8403 . . . . 5 (𝐴𝑉 → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴 → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≼ 𝐴))
4948adantr 481 . . . 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 6084 . . . . 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 1126 . . . . . 6 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On)
54 elpw2g 5138 . . . . . . . . . 10 (𝐴𝑉 → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴))
5554adantr 481 . . . . . . . . 9 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ⊆ 𝐴))
5647, 55mpbird 258 . . . . . . . 8 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴)
57563adant3 1125 . . . . . . 7 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴)
58 simp3 1131 . . . . . . 7 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴)
59 nfcv 2949 . . . . . . . . 9 𝑥On
60 nfrab1 3344 . . . . . . . . 9 𝑥{𝑥 ∈ 𝒫 𝐴𝑥𝐴}
6159, 60nfin 4113 . . . . . . . 8 𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
62 nfcv 2949 . . . . . . . 8 𝑥𝒫 𝐴
63 nfcv 2949 . . . . . . . . 9 𝑥
64 nfcv 2949 . . . . . . . . 9 𝑥𝐴
6561, 63, 64nfbr 5009 . . . . . . . 8 𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴
66 breq1 4965 . . . . . . . 8 (𝑥 = (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) → (𝑥𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
6761, 62, 65, 66elrabf 3614 . . . . . . 7 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ↔ ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ 𝒫 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
6857, 58, 67sylanbrc 583 . . . . . 6 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})
6953, 68elind 4092 . . . . 5 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}))
70693expia 1114 . . . 4 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴 → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴})))
7152, 70mtod 199 . . 3 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴)
72 bren2 8388 . . 3 ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴 ↔ ((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≼ 𝐴 ∧ ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≺ 𝐴))
7350, 71, 72sylanbrc 583 . 2 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴)
74 isnumi 9221 . 2 (((On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ∈ On ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴𝑥𝐴}) ≈ 𝐴) → 𝐴 ∈ dom card)
7545, 73, 74syl2anc 584 1 ((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → 𝐴 ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1080  wal 1520  wcel 2081  {crab 3109  Vcvv 3437  cin 3858  wss 3859  𝒫 cpw 4453   class class class wbr 4962  Tr wtr 5063  dom cdm 5443  Ord word 6065  Oncon0 6066  cen 8354  cdom 8355  csdm 8356  cardccrd 9210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-ord 6069  df-on 6070  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-card 9214
This theorem is referenced by:  tskwe2  10041  grothac  10098
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