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Theorem cardprclem 9980
Description: Lemma for cardprc 9981. (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 2824 . . . . . . . 8 (𝑥𝐴𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥})
3 abid 2712 . . . . . . . 8 (𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥} ↔ (card‘𝑥) = 𝑥)
4 iscard 9976 . . . . . . . 8 ((card‘𝑥) = 𝑥 ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 𝑦𝑥))
52, 3, 43bitri 297 . . . . . . 7 (𝑥𝐴 ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 𝑦𝑥))
65simplbi 497 . . . . . 6 (𝑥𝐴𝑥 ∈ On)
76ssriv 3986 . . . . 5 𝐴 ⊆ On
8 ssonuni 7771 . . . . 5 (𝐴 ∈ V → (𝐴 ⊆ On → 𝐴 ∈ On))
97, 8mpi 20 . . . 4 (𝐴 ∈ V → 𝐴 ∈ On)
10 domrefg 8989 . . . . 5 ( 𝐴 ∈ On → 𝐴 𝐴)
119, 10syl 17 . . . 4 (𝐴 ∈ V → 𝐴 𝐴)
12 elharval 9562 . . . 4 ( 𝐴 ∈ (har‘ 𝐴) ↔ ( 𝐴 ∈ On ∧ 𝐴 𝐴))
139, 11, 12sylanbrc 582 . . 3 (𝐴 ∈ V → 𝐴 ∈ (har‘ 𝐴))
147sseli 3978 . . . . . . . 8 (𝑧𝐴𝑧 ∈ On)
15 domrefg 8989 . . . . . . . . . 10 (𝑧 ∈ On → 𝑧𝑧)
1615ancli 548 . . . . . . . . 9 (𝑧 ∈ On → (𝑧 ∈ On ∧ 𝑧𝑧))
17 elharval 9562 . . . . . . . . 9 (𝑧 ∈ (har‘𝑧) ↔ (𝑧 ∈ On ∧ 𝑧𝑧))
1816, 17sylibr 233 . . . . . . . 8 (𝑧 ∈ On → 𝑧 ∈ (har‘𝑧))
1914, 18syl 17 . . . . . . 7 (𝑧𝐴𝑧 ∈ (har‘𝑧))
20 harcard 9979 . . . . . . . 8 (card‘(har‘𝑧)) = (har‘𝑧)
21 fvex 6904 . . . . . . . . 9 (har‘𝑧) ∈ V
22 fveq2 6891 . . . . . . . . . 10 (𝑥 = (har‘𝑧) → (card‘𝑥) = (card‘(har‘𝑧)))
23 id 22 . . . . . . . . . 10 (𝑥 = (har‘𝑧) → 𝑥 = (har‘𝑧))
2422, 23eqeq12d 2747 . . . . . . . . 9 (𝑥 = (har‘𝑧) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘𝑧)) = (har‘𝑧)))
2521, 24, 1elab2 3672 . . . . . . . 8 ((har‘𝑧) ∈ 𝐴 ↔ (card‘(har‘𝑧)) = (har‘𝑧))
2620, 25mpbir 230 . . . . . . 7 (har‘𝑧) ∈ 𝐴
27 eleq2 2821 . . . . . . . . 9 (𝑤 = (har‘𝑧) → (𝑧𝑤𝑧 ∈ (har‘𝑧)))
28 eleq1 2820 . . . . . . . . 9 (𝑤 = (har‘𝑧) → (𝑤𝐴 ↔ (har‘𝑧) ∈ 𝐴))
2927, 28anbi12d 630 . . . . . . . 8 (𝑤 = (har‘𝑧) → ((𝑧𝑤𝑤𝐴) ↔ (𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴)))
3021, 29spcev 3596 . . . . . . 7 ((𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴) → ∃𝑤(𝑧𝑤𝑤𝐴))
3119, 26, 30sylancl 585 . . . . . 6 (𝑧𝐴 → ∃𝑤(𝑧𝑤𝑤𝐴))
32 eluni 4911 . . . . . 6 (𝑧 𝐴 ↔ ∃𝑤(𝑧𝑤𝑤𝐴))
3331, 32sylibr 233 . . . . 5 (𝑧𝐴𝑧 𝐴)
3433ssriv 3986 . . . 4 𝐴 𝐴
35 harcard 9979 . . . . 5 (card‘(har‘ 𝐴)) = (har‘ 𝐴)
36 fvex 6904 . . . . . 6 (har‘ 𝐴) ∈ V
37 fveq2 6891 . . . . . . 7 (𝑥 = (har‘ 𝐴) → (card‘𝑥) = (card‘(har‘ 𝐴)))
38 id 22 . . . . . . 7 (𝑥 = (har‘ 𝐴) → 𝑥 = (har‘ 𝐴))
3937, 38eqeq12d 2747 . . . . . 6 (𝑥 = (har‘ 𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘ 𝐴)) = (har‘ 𝐴)))
4036, 39, 1elab2 3672 . . . . 5 ((har‘ 𝐴) ∈ 𝐴 ↔ (card‘(har‘ 𝐴)) = (har‘ 𝐴))
4135, 40mpbir 230 . . . 4 (har‘ 𝐴) ∈ 𝐴
4234, 41sselii 3979 . . 3 (har‘ 𝐴) ∈ 𝐴
4313, 42jctir 520 . 2 (𝐴 ∈ V → ( 𝐴 ∈ (har‘ 𝐴) ∧ (har‘ 𝐴) ∈ 𝐴))
44 eloni 6374 . . 3 ( 𝐴 ∈ On → Ord 𝐴)
45 ordn2lp 6384 . . 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 193 1 ¬ 𝐴 ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1540  wex 1780  wcel 2105  {cab 2708  wral 3060  Vcvv 3473  wss 3948   cuni 4908   class class class wbr 5148  Ord word 6363  Oncon0 6364  cfv 6543  cdom 8943  csdm 8944  harchar 9557  cardccrd 9936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-oi 9511  df-har 9558  df-card 9940
This theorem is referenced by:  cardprc  9981
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