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Theorem cardprclem 9483
Description: Lemma for cardprc 9484. (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 2720 . . . . . . . 8 (𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥} ↔ (card‘𝑥) = 𝑥)
4 iscard 9479 . . . . . . . 8 ((card‘𝑥) = 𝑥 ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 𝑦𝑥))
52, 3, 43bitri 300 . . . . . . 7 (𝑥𝐴 ↔ (𝑥 ∈ On ∧ ∀𝑦𝑥 𝑦𝑥))
65simplbi 501 . . . . . 6 (𝑥𝐴𝑥 ∈ On)
76ssriv 3881 . . . . 5 𝐴 ⊆ On
8 ssonuni 7522 . . . . 5 (𝐴 ∈ V → (𝐴 ⊆ On → 𝐴 ∈ On))
97, 8mpi 20 . . . 4 (𝐴 ∈ V → 𝐴 ∈ On)
10 domrefg 8592 . . . . 5 ( 𝐴 ∈ On → 𝐴 𝐴)
119, 10syl 17 . . . 4 (𝐴 ∈ V → 𝐴 𝐴)
12 elharval 9100 . . . 4 ( 𝐴 ∈ (har‘ 𝐴) ↔ ( 𝐴 ∈ On ∧ 𝐴 𝐴))
139, 11, 12sylanbrc 586 . . 3 (𝐴 ∈ V → 𝐴 ∈ (har‘ 𝐴))
147sseli 3873 . . . . . . . 8 (𝑧𝐴𝑧 ∈ On)
15 domrefg 8592 . . . . . . . . . 10 (𝑧 ∈ On → 𝑧𝑧)
1615ancli 552 . . . . . . . . 9 (𝑧 ∈ On → (𝑧 ∈ On ∧ 𝑧𝑧))
17 elharval 9100 . . . . . . . . 9 (𝑧 ∈ (har‘𝑧) ↔ (𝑧 ∈ On ∧ 𝑧𝑧))
1816, 17sylibr 237 . . . . . . . 8 (𝑧 ∈ On → 𝑧 ∈ (har‘𝑧))
1914, 18syl 17 . . . . . . 7 (𝑧𝐴𝑧 ∈ (har‘𝑧))
20 harcard 9482 . . . . . . . 8 (card‘(har‘𝑧)) = (har‘𝑧)
21 fvex 6689 . . . . . . . . 9 (har‘𝑧) ∈ V
22 fveq2 6676 . . . . . . . . . 10 (𝑥 = (har‘𝑧) → (card‘𝑥) = (card‘(har‘𝑧)))
23 id 22 . . . . . . . . . 10 (𝑥 = (har‘𝑧) → 𝑥 = (har‘𝑧))
2422, 23eqeq12d 2754 . . . . . . . . 9 (𝑥 = (har‘𝑧) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘𝑧)) = (har‘𝑧)))
2521, 24, 1elab2 3577 . . . . . . . 8 ((har‘𝑧) ∈ 𝐴 ↔ (card‘(har‘𝑧)) = (har‘𝑧))
2620, 25mpbir 234 . . . . . . 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 634 . . . . . . . 8 (𝑤 = (har‘𝑧) → ((𝑧𝑤𝑤𝐴) ↔ (𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴)))
3021, 29spcev 3510 . . . . . . 7 ((𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴) → ∃𝑤(𝑧𝑤𝑤𝐴))
3119, 26, 30sylancl 589 . . . . . 6 (𝑧𝐴 → ∃𝑤(𝑧𝑤𝑤𝐴))
32 eluni 4799 . . . . . 6 (𝑧 𝐴 ↔ ∃𝑤(𝑧𝑤𝑤𝐴))
3331, 32sylibr 237 . . . . 5 (𝑧𝐴𝑧 𝐴)
3433ssriv 3881 . . . 4 𝐴 𝐴
35 harcard 9482 . . . . 5 (card‘(har‘ 𝐴)) = (har‘ 𝐴)
36 fvex 6689 . . . . . 6 (har‘ 𝐴) ∈ V
37 fveq2 6676 . . . . . . 7 (𝑥 = (har‘ 𝐴) → (card‘𝑥) = (card‘(har‘ 𝐴)))
38 id 22 . . . . . . 7 (𝑥 = (har‘ 𝐴) → 𝑥 = (har‘ 𝐴))
3937, 38eqeq12d 2754 . . . . . 6 (𝑥 = (har‘ 𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘ 𝐴)) = (har‘ 𝐴)))
4036, 39, 1elab2 3577 . . . . 5 ((har‘ 𝐴) ∈ 𝐴 ↔ (card‘(har‘ 𝐴)) = (har‘ 𝐴))
4135, 40mpbir 234 . . . 4 (har‘ 𝐴) ∈ 𝐴
4234, 41sselii 3874 . . 3 (har‘ 𝐴) ∈ 𝐴
4313, 42jctir 524 . 2 (𝐴 ∈ V → ( 𝐴 ∈ (har‘ 𝐴) ∧ (har‘ 𝐴) ∈ 𝐴))
44 eloni 6182 . . 3 ( 𝐴 ∈ On → Ord 𝐴)
45 ordn2lp 6192 . . 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 197 1 ¬ 𝐴 ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1542  wex 1786  wcel 2114  {cab 2716  wral 3053  Vcvv 3398  wss 3843   cuni 4796   class class class wbr 5030  Ord word 6171  Oncon0 6172  cfv 6339  cdom 8555  csdm 8556  harchar 9095  cardccrd 9439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7129  df-wrecs 7978  df-recs 8039  df-er 8322  df-en 8558  df-dom 8559  df-sdom 8560  df-oi 9049  df-har 9096  df-card 9443
This theorem is referenced by:  cardprc  9484
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