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Theorem cardprclem 8688
Description: Lemma for cardprc 8689. (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 2680 . . . . . . . 8 (𝑥𝐴𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥})
3 abid 2598 . . . . . . . 8 (𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥} ↔ (card‘𝑥) = 𝑥)
4 iscard 8684 . . . . . . . 8 ((card‘𝑥) = 𝑥 ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 𝑦𝑥))
52, 3, 43bitri 285 . . . . . . 7 (𝑥𝐴 ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 𝑦𝑥))
65simplbi 475 . . . . . 6 (𝑥𝐴𝑥 ∈ On)
76ssriv 3572 . . . . 5 𝐴 ⊆ On
8 ssonuni 6878 . . . . 5 (𝐴 ∈ V → (𝐴 ⊆ On → 𝐴 ∈ On))
97, 8mpi 20 . . . 4 (𝐴 ∈ V → 𝐴 ∈ On)
10 domrefg 7876 . . . . 5 ( 𝐴 ∈ On → 𝐴 𝐴)
119, 10syl 17 . . . 4 (𝐴 ∈ V → 𝐴 𝐴)
12 elharval 8351 . . . 4 ( 𝐴 ∈ (har‘ 𝐴) ↔ ( 𝐴 ∈ On ∧ 𝐴 𝐴))
139, 11, 12sylanbrc 695 . . 3 (𝐴 ∈ V → 𝐴 ∈ (har‘ 𝐴))
147sseli 3564 . . . . . . . 8 (𝑧𝐴𝑧 ∈ On)
15 domrefg 7876 . . . . . . . . . 10 (𝑧 ∈ On → 𝑧𝑧)
1615ancli 572 . . . . . . . . 9 (𝑧 ∈ On → (𝑧 ∈ On ∧ 𝑧𝑧))
17 elharval 8351 . . . . . . . . 9 (𝑧 ∈ (har‘𝑧) ↔ (𝑧 ∈ On ∧ 𝑧𝑧))
1816, 17sylibr 223 . . . . . . . 8 (𝑧 ∈ On → 𝑧 ∈ (har‘𝑧))
1914, 18syl 17 . . . . . . 7 (𝑧𝐴𝑧 ∈ (har‘𝑧))
20 harcard 8687 . . . . . . . 8 (card‘(har‘𝑧)) = (har‘𝑧)
21 fvex 6113 . . . . . . . . 9 (har‘𝑧) ∈ V
22 fveq2 6103 . . . . . . . . . 10 (𝑥 = (har‘𝑧) → (card‘𝑥) = (card‘(har‘𝑧)))
23 id 22 . . . . . . . . . 10 (𝑥 = (har‘𝑧) → 𝑥 = (har‘𝑧))
2422, 23eqeq12d 2625 . . . . . . . . 9 (𝑥 = (har‘𝑧) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘𝑧)) = (har‘𝑧)))
2521, 24, 1elab2 3323 . . . . . . . 8 ((har‘𝑧) ∈ 𝐴 ↔ (card‘(har‘𝑧)) = (har‘𝑧))
2620, 25mpbir 220 . . . . . . 7 (har‘𝑧) ∈ 𝐴
27 eleq2 2677 . . . . . . . . 9 (𝑤 = (har‘𝑧) → (𝑧𝑤𝑧 ∈ (har‘𝑧)))
28 eleq1 2676 . . . . . . . . 9 (𝑤 = (har‘𝑧) → (𝑤𝐴 ↔ (har‘𝑧) ∈ 𝐴))
2927, 28anbi12d 743 . . . . . . . 8 (𝑤 = (har‘𝑧) → ((𝑧𝑤𝑤𝐴) ↔ (𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴)))
3021, 29spcev 3273 . . . . . . 7 ((𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴) → ∃𝑤(𝑧𝑤𝑤𝐴))
3119, 26, 30sylancl 693 . . . . . 6 (𝑧𝐴 → ∃𝑤(𝑧𝑤𝑤𝐴))
32 eluni 4375 . . . . . 6 (𝑧 𝐴 ↔ ∃𝑤(𝑧𝑤𝑤𝐴))
3331, 32sylibr 223 . . . . 5 (𝑧𝐴𝑧 𝐴)
3433ssriv 3572 . . . 4 𝐴 𝐴
35 harcard 8687 . . . . 5 (card‘(har‘ 𝐴)) = (har‘ 𝐴)
36 fvex 6113 . . . . . 6 (har‘ 𝐴) ∈ V
37 fveq2 6103 . . . . . . 7 (𝑥 = (har‘ 𝐴) → (card‘𝑥) = (card‘(har‘ 𝐴)))
38 id 22 . . . . . . 7 (𝑥 = (har‘ 𝐴) → 𝑥 = (har‘ 𝐴))
3937, 38eqeq12d 2625 . . . . . 6 (𝑥 = (har‘ 𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘ 𝐴)) = (har‘ 𝐴)))
4036, 39, 1elab2 3323 . . . . 5 ((har‘ 𝐴) ∈ 𝐴 ↔ (card‘(har‘ 𝐴)) = (har‘ 𝐴))
4135, 40mpbir 220 . . . 4 (har‘ 𝐴) ∈ 𝐴
4234, 41sselii 3565 . . 3 (har‘ 𝐴) ∈ 𝐴
4313, 42jctir 559 . 2 (𝐴 ∈ V → ( 𝐴 ∈ (har‘ 𝐴) ∧ (har‘ 𝐴) ∈ 𝐴))
44 eloni 5650 . . 3 ( 𝐴 ∈ On → Ord 𝐴)
45 ordn2lp 5660 . . 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 184 1 ¬ 𝐴 ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1475  wex 1695  wcel 1977  {cab 2596  wral 2896  Vcvv 3173  wss 3540   cuni 4372   class class class wbr 4583  Ord word 5639  Oncon0 5640  cfv 5804  cdom 7839  csdm 7840  harchar 8344  cardccrd 8644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-wrecs 7294  df-recs 7355  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-oi 8298  df-har 8346  df-card 8648
This theorem is referenced by:  cardprc  8689
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