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Theorem cardprclem 8752
Description: Lemma for cardprc 8753. (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 2690 . . . . . . . 8 (𝑥𝐴𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥})
3 abid 2609 . . . . . . . 8 (𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥} ↔ (card‘𝑥) = 𝑥)
4 iscard 8748 . . . . . . . 8 ((card‘𝑥) = 𝑥 ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 𝑦𝑥))
52, 3, 43bitri 286 . . . . . . 7 (𝑥𝐴 ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 𝑦𝑥))
65simplbi 476 . . . . . 6 (𝑥𝐴𝑥 ∈ On)
76ssriv 3588 . . . . 5 𝐴 ⊆ On
8 ssonuni 6936 . . . . 5 (𝐴 ∈ V → (𝐴 ⊆ On → 𝐴 ∈ On))
97, 8mpi 20 . . . 4 (𝐴 ∈ V → 𝐴 ∈ On)
10 domrefg 7937 . . . . 5 ( 𝐴 ∈ On → 𝐴 𝐴)
119, 10syl 17 . . . 4 (𝐴 ∈ V → 𝐴 𝐴)
12 elharval 8415 . . . 4 ( 𝐴 ∈ (har‘ 𝐴) ↔ ( 𝐴 ∈ On ∧ 𝐴 𝐴))
139, 11, 12sylanbrc 697 . . 3 (𝐴 ∈ V → 𝐴 ∈ (har‘ 𝐴))
147sseli 3580 . . . . . . . 8 (𝑧𝐴𝑧 ∈ On)
15 domrefg 7937 . . . . . . . . . 10 (𝑧 ∈ On → 𝑧𝑧)
1615ancli 573 . . . . . . . . 9 (𝑧 ∈ On → (𝑧 ∈ On ∧ 𝑧𝑧))
17 elharval 8415 . . . . . . . . 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 8751 . . . . . . . 8 (card‘(har‘𝑧)) = (har‘𝑧)
21 fvex 6160 . . . . . . . . 9 (har‘𝑧) ∈ V
22 fveq2 6150 . . . . . . . . . 10 (𝑥 = (har‘𝑧) → (card‘𝑥) = (card‘(har‘𝑧)))
23 id 22 . . . . . . . . . 10 (𝑥 = (har‘𝑧) → 𝑥 = (har‘𝑧))
2422, 23eqeq12d 2636 . . . . . . . . 9 (𝑥 = (har‘𝑧) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘𝑧)) = (har‘𝑧)))
2521, 24, 1elab2 3338 . . . . . . . 8 ((har‘𝑧) ∈ 𝐴 ↔ (card‘(har‘𝑧)) = (har‘𝑧))
2620, 25mpbir 221 . . . . . . 7 (har‘𝑧) ∈ 𝐴
27 eleq2 2687 . . . . . . . . 9 (𝑤 = (har‘𝑧) → (𝑧𝑤𝑧 ∈ (har‘𝑧)))
28 eleq1 2686 . . . . . . . . 9 (𝑤 = (har‘𝑧) → (𝑤𝐴 ↔ (har‘𝑧) ∈ 𝐴))
2927, 28anbi12d 746 . . . . . . . 8 (𝑤 = (har‘𝑧) → ((𝑧𝑤𝑤𝐴) ↔ (𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴)))
3021, 29spcev 3286 . . . . . . 7 ((𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴) → ∃𝑤(𝑧𝑤𝑤𝐴))
3119, 26, 30sylancl 693 . . . . . 6 (𝑧𝐴 → ∃𝑤(𝑧𝑤𝑤𝐴))
32 eluni 4407 . . . . . 6 (𝑧 𝐴 ↔ ∃𝑤(𝑧𝑤𝑤𝐴))
3331, 32sylibr 224 . . . . 5 (𝑧𝐴𝑧 𝐴)
3433ssriv 3588 . . . 4 𝐴 𝐴
35 harcard 8751 . . . . 5 (card‘(har‘ 𝐴)) = (har‘ 𝐴)
36 fvex 6160 . . . . . 6 (har‘ 𝐴) ∈ V
37 fveq2 6150 . . . . . . 7 (𝑥 = (har‘ 𝐴) → (card‘𝑥) = (card‘(har‘ 𝐴)))
38 id 22 . . . . . . 7 (𝑥 = (har‘ 𝐴) → 𝑥 = (har‘ 𝐴))
3937, 38eqeq12d 2636 . . . . . 6 (𝑥 = (har‘ 𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘ 𝐴)) = (har‘ 𝐴)))
4036, 39, 1elab2 3338 . . . . 5 ((har‘ 𝐴) ∈ 𝐴 ↔ (card‘(har‘ 𝐴)) = (har‘ 𝐴))
4135, 40mpbir 221 . . . 4 (har‘ 𝐴) ∈ 𝐴
4234, 41sselii 3581 . . 3 (har‘ 𝐴) ∈ 𝐴
4313, 42jctir 560 . 2 (𝐴 ∈ V → ( 𝐴 ∈ (har‘ 𝐴) ∧ (har‘ 𝐴) ∈ 𝐴))
44 eloni 5694 . . 3 ( 𝐴 ∈ On → Ord 𝐴)
45 ordn2lp 5704 . . 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 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  Vcvv 3186  wss 3556   cuni 4404   class class class wbr 4615  Ord word 5683  Oncon0 5684  cfv 5849  cdom 7900  csdm 7901  harchar 8408  cardccrd 8708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-wrecs 7355  df-recs 7416  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-oi 8362  df-har 8410  df-card 8712
This theorem is referenced by:  cardprc  8753
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