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Theorem cardprclem 8843
Description: Lemma for cardprc 8844. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
Hypothesis
Ref Expression
cardprclem.1 𝐴 = {𝑥 ∣ (card‘𝑥) = 𝑥}
Assertion
Ref Expression
cardprclem ¬ 𝐴 ∈ V
Distinct variable group:   𝑥,𝐴

Proof of Theorem cardprclem
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardprclem.1 . . . . . . . . 9 𝐴 = {𝑥 ∣ (card‘𝑥) = 𝑥}
21eleq2i 2722 . . . . . . . 8 (𝑥𝐴𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥})
3 abid 2639 . . . . . . . 8 (𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥} ↔ (card‘𝑥) = 𝑥)
4 iscard 8839 . . . . . . . 8 ((card‘𝑥) = 𝑥 ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 𝑦𝑥))
52, 3, 43bitri 286 . . . . . . 7 (𝑥𝐴 ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 𝑦𝑥))
65simplbi 475 . . . . . 6 (𝑥𝐴𝑥 ∈ On)
76ssriv 3640 . . . . 5 𝐴 ⊆ On
8 ssonuni 7028 . . . . 5 (𝐴 ∈ V → (𝐴 ⊆ On → 𝐴 ∈ On))
97, 8mpi 20 . . . 4 (𝐴 ∈ V → 𝐴 ∈ On)
10 domrefg 8032 . . . . 5 ( 𝐴 ∈ On → 𝐴 𝐴)
119, 10syl 17 . . . 4 (𝐴 ∈ V → 𝐴 𝐴)
12 elharval 8509 . . . 4 ( 𝐴 ∈ (har‘ 𝐴) ↔ ( 𝐴 ∈ On ∧ 𝐴 𝐴))
139, 11, 12sylanbrc 699 . . 3 (𝐴 ∈ V → 𝐴 ∈ (har‘ 𝐴))
147sseli 3632 . . . . . . . 8 (𝑧𝐴𝑧 ∈ On)
15 domrefg 8032 . . . . . . . . . 10 (𝑧 ∈ On → 𝑧𝑧)
1615ancli 573 . . . . . . . . 9 (𝑧 ∈ On → (𝑧 ∈ On ∧ 𝑧𝑧))
17 elharval 8509 . . . . . . . . 9 (𝑧 ∈ (har‘𝑧) ↔ (𝑧 ∈ On ∧ 𝑧𝑧))
1816, 17sylibr 224 . . . . . . . 8 (𝑧 ∈ On → 𝑧 ∈ (har‘𝑧))
1914, 18syl 17 . . . . . . 7 (𝑧𝐴𝑧 ∈ (har‘𝑧))
20 harcard 8842 . . . . . . . 8 (card‘(har‘𝑧)) = (har‘𝑧)
21 fvex 6239 . . . . . . . . 9 (har‘𝑧) ∈ V
22 fveq2 6229 . . . . . . . . . 10 (𝑥 = (har‘𝑧) → (card‘𝑥) = (card‘(har‘𝑧)))
23 id 22 . . . . . . . . . 10 (𝑥 = (har‘𝑧) → 𝑥 = (har‘𝑧))
2422, 23eqeq12d 2666 . . . . . . . . 9 (𝑥 = (har‘𝑧) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘𝑧)) = (har‘𝑧)))
2521, 24, 1elab2 3386 . . . . . . . 8 ((har‘𝑧) ∈ 𝐴 ↔ (card‘(har‘𝑧)) = (har‘𝑧))
2620, 25mpbir 221 . . . . . . 7 (har‘𝑧) ∈ 𝐴
27 eleq2 2719 . . . . . . . . 9 (𝑤 = (har‘𝑧) → (𝑧𝑤𝑧 ∈ (har‘𝑧)))
28 eleq1 2718 . . . . . . . . 9 (𝑤 = (har‘𝑧) → (𝑤𝐴 ↔ (har‘𝑧) ∈ 𝐴))
2927, 28anbi12d 747 . . . . . . . 8 (𝑤 = (har‘𝑧) → ((𝑧𝑤𝑤𝐴) ↔ (𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴)))
3021, 29spcev 3331 . . . . . . 7 ((𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴) → ∃𝑤(𝑧𝑤𝑤𝐴))
3119, 26, 30sylancl 695 . . . . . 6 (𝑧𝐴 → ∃𝑤(𝑧𝑤𝑤𝐴))
32 eluni 4471 . . . . . 6 (𝑧 𝐴 ↔ ∃𝑤(𝑧𝑤𝑤𝐴))
3331, 32sylibr 224 . . . . 5 (𝑧𝐴𝑧 𝐴)
3433ssriv 3640 . . . 4 𝐴 𝐴
35 harcard 8842 . . . . 5 (card‘(har‘ 𝐴)) = (har‘ 𝐴)
36 fvex 6239 . . . . . 6 (har‘ 𝐴) ∈ V
37 fveq2 6229 . . . . . . 7 (𝑥 = (har‘ 𝐴) → (card‘𝑥) = (card‘(har‘ 𝐴)))
38 id 22 . . . . . . 7 (𝑥 = (har‘ 𝐴) → 𝑥 = (har‘ 𝐴))
3937, 38eqeq12d 2666 . . . . . 6 (𝑥 = (har‘ 𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘ 𝐴)) = (har‘ 𝐴)))
4036, 39, 1elab2 3386 . . . . 5 ((har‘ 𝐴) ∈ 𝐴 ↔ (card‘(har‘ 𝐴)) = (har‘ 𝐴))
4135, 40mpbir 221 . . . 4 (har‘ 𝐴) ∈ 𝐴
4234, 41sselii 3633 . . 3 (har‘ 𝐴) ∈ 𝐴
4313, 42jctir 560 . 2 (𝐴 ∈ V → ( 𝐴 ∈ (har‘ 𝐴) ∧ (har‘ 𝐴) ∈ 𝐴))
44 eloni 5771 . . 3 ( 𝐴 ∈ On → Ord 𝐴)
45 ordn2lp 5781 . . 3 (Ord 𝐴 → ¬ ( 𝐴 ∈ (har‘ 𝐴) ∧ (har‘ 𝐴) ∈ 𝐴))
469, 44, 453syl 18 . 2 (𝐴 ∈ V → ¬ ( 𝐴 ∈ (har‘ 𝐴) ∧ (har‘ 𝐴) ∈ 𝐴))
4743, 46pm2.65i 185 1 ¬ 𝐴 ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  Vcvv 3231  wss 3607   cuni 4468   class class class wbr 4685  Ord word 5760  Oncon0 5761  cfv 5926  cdom 7995  csdm 7996  harchar 8502  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-rep 4804  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-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  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-iun 4554  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-se 5103  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-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-wrecs 7452  df-recs 7513  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-oi 8456  df-har 8504  df-card 8803
This theorem is referenced by:  cardprc  8844
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