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Theorem cardprclem 9920
Description: Lemma for cardprc 9921. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
Hypothesis
Ref Expression
cardprclem.1 𝐴 = {π‘₯ ∣ (cardβ€˜π‘₯) = π‘₯}
Assertion
Ref Expression
cardprclem ¬ 𝐴 ∈ V
Distinct variable group:   π‘₯,𝐴

Proof of Theorem cardprclem
Dummy variables 𝑀 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardprclem.1 . . . . . . . . 9 𝐴 = {π‘₯ ∣ (cardβ€˜π‘₯) = π‘₯}
21eleq2i 2826 . . . . . . . 8 (π‘₯ ∈ 𝐴 ↔ π‘₯ ∈ {π‘₯ ∣ (cardβ€˜π‘₯) = π‘₯})
3 abid 2714 . . . . . . . 8 (π‘₯ ∈ {π‘₯ ∣ (cardβ€˜π‘₯) = π‘₯} ↔ (cardβ€˜π‘₯) = π‘₯)
4 iscard 9916 . . . . . . . 8 ((cardβ€˜π‘₯) = π‘₯ ↔ (π‘₯ ∈ On ∧ βˆ€π‘¦ ∈ π‘₯ 𝑦 β‰Ί π‘₯))
52, 3, 43bitri 297 . . . . . . 7 (π‘₯ ∈ 𝐴 ↔ (π‘₯ ∈ On ∧ βˆ€π‘¦ ∈ π‘₯ 𝑦 β‰Ί π‘₯))
65simplbi 499 . . . . . 6 (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ On)
76ssriv 3949 . . . . 5 𝐴 βŠ† On
8 ssonuni 7715 . . . . 5 (𝐴 ∈ V β†’ (𝐴 βŠ† On β†’ βˆͺ 𝐴 ∈ On))
97, 8mpi 20 . . . 4 (𝐴 ∈ V β†’ βˆͺ 𝐴 ∈ On)
10 domrefg 8930 . . . . 5 (βˆͺ 𝐴 ∈ On β†’ βˆͺ 𝐴 β‰Ό βˆͺ 𝐴)
119, 10syl 17 . . . 4 (𝐴 ∈ V β†’ βˆͺ 𝐴 β‰Ό βˆͺ 𝐴)
12 elharval 9502 . . . 4 (βˆͺ 𝐴 ∈ (harβ€˜βˆͺ 𝐴) ↔ (βˆͺ 𝐴 ∈ On ∧ βˆͺ 𝐴 β‰Ό βˆͺ 𝐴))
139, 11, 12sylanbrc 584 . . 3 (𝐴 ∈ V β†’ βˆͺ 𝐴 ∈ (harβ€˜βˆͺ 𝐴))
147sseli 3941 . . . . . . . 8 (𝑧 ∈ 𝐴 β†’ 𝑧 ∈ On)
15 domrefg 8930 . . . . . . . . . 10 (𝑧 ∈ On β†’ 𝑧 β‰Ό 𝑧)
1615ancli 550 . . . . . . . . 9 (𝑧 ∈ On β†’ (𝑧 ∈ On ∧ 𝑧 β‰Ό 𝑧))
17 elharval 9502 . . . . . . . . 9 (𝑧 ∈ (harβ€˜π‘§) ↔ (𝑧 ∈ On ∧ 𝑧 β‰Ό 𝑧))
1816, 17sylibr 233 . . . . . . . 8 (𝑧 ∈ On β†’ 𝑧 ∈ (harβ€˜π‘§))
1914, 18syl 17 . . . . . . 7 (𝑧 ∈ 𝐴 β†’ 𝑧 ∈ (harβ€˜π‘§))
20 harcard 9919 . . . . . . . 8 (cardβ€˜(harβ€˜π‘§)) = (harβ€˜π‘§)
21 fvex 6856 . . . . . . . . 9 (harβ€˜π‘§) ∈ V
22 fveq2 6843 . . . . . . . . . 10 (π‘₯ = (harβ€˜π‘§) β†’ (cardβ€˜π‘₯) = (cardβ€˜(harβ€˜π‘§)))
23 id 22 . . . . . . . . . 10 (π‘₯ = (harβ€˜π‘§) β†’ π‘₯ = (harβ€˜π‘§))
2422, 23eqeq12d 2749 . . . . . . . . 9 (π‘₯ = (harβ€˜π‘§) β†’ ((cardβ€˜π‘₯) = π‘₯ ↔ (cardβ€˜(harβ€˜π‘§)) = (harβ€˜π‘§)))
2521, 24, 1elab2 3635 . . . . . . . 8 ((harβ€˜π‘§) ∈ 𝐴 ↔ (cardβ€˜(harβ€˜π‘§)) = (harβ€˜π‘§))
2620, 25mpbir 230 . . . . . . 7 (harβ€˜π‘§) ∈ 𝐴
27 eleq2 2823 . . . . . . . . 9 (𝑀 = (harβ€˜π‘§) β†’ (𝑧 ∈ 𝑀 ↔ 𝑧 ∈ (harβ€˜π‘§)))
28 eleq1 2822 . . . . . . . . 9 (𝑀 = (harβ€˜π‘§) β†’ (𝑀 ∈ 𝐴 ↔ (harβ€˜π‘§) ∈ 𝐴))
2927, 28anbi12d 632 . . . . . . . 8 (𝑀 = (harβ€˜π‘§) β†’ ((𝑧 ∈ 𝑀 ∧ 𝑀 ∈ 𝐴) ↔ (𝑧 ∈ (harβ€˜π‘§) ∧ (harβ€˜π‘§) ∈ 𝐴)))
3021, 29spcev 3564 . . . . . . 7 ((𝑧 ∈ (harβ€˜π‘§) ∧ (harβ€˜π‘§) ∈ 𝐴) β†’ βˆƒπ‘€(𝑧 ∈ 𝑀 ∧ 𝑀 ∈ 𝐴))
3119, 26, 30sylancl 587 . . . . . 6 (𝑧 ∈ 𝐴 β†’ βˆƒπ‘€(𝑧 ∈ 𝑀 ∧ 𝑀 ∈ 𝐴))
32 eluni 4869 . . . . . 6 (𝑧 ∈ βˆͺ 𝐴 ↔ βˆƒπ‘€(𝑧 ∈ 𝑀 ∧ 𝑀 ∈ 𝐴))
3331, 32sylibr 233 . . . . 5 (𝑧 ∈ 𝐴 β†’ 𝑧 ∈ βˆͺ 𝐴)
3433ssriv 3949 . . . 4 𝐴 βŠ† βˆͺ 𝐴
35 harcard 9919 . . . . 5 (cardβ€˜(harβ€˜βˆͺ 𝐴)) = (harβ€˜βˆͺ 𝐴)
36 fvex 6856 . . . . . 6 (harβ€˜βˆͺ 𝐴) ∈ V
37 fveq2 6843 . . . . . . 7 (π‘₯ = (harβ€˜βˆͺ 𝐴) β†’ (cardβ€˜π‘₯) = (cardβ€˜(harβ€˜βˆͺ 𝐴)))
38 id 22 . . . . . . 7 (π‘₯ = (harβ€˜βˆͺ 𝐴) β†’ π‘₯ = (harβ€˜βˆͺ 𝐴))
3937, 38eqeq12d 2749 . . . . . 6 (π‘₯ = (harβ€˜βˆͺ 𝐴) β†’ ((cardβ€˜π‘₯) = π‘₯ ↔ (cardβ€˜(harβ€˜βˆͺ 𝐴)) = (harβ€˜βˆͺ 𝐴)))
4036, 39, 1elab2 3635 . . . . 5 ((harβ€˜βˆͺ 𝐴) ∈ 𝐴 ↔ (cardβ€˜(harβ€˜βˆͺ 𝐴)) = (harβ€˜βˆͺ 𝐴))
4135, 40mpbir 230 . . . 4 (harβ€˜βˆͺ 𝐴) ∈ 𝐴
4234, 41sselii 3942 . . 3 (harβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴
4313, 42jctir 522 . 2 (𝐴 ∈ V β†’ (βˆͺ 𝐴 ∈ (harβ€˜βˆͺ 𝐴) ∧ (harβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴))
44 eloni 6328 . . 3 (βˆͺ 𝐴 ∈ On β†’ Ord βˆͺ 𝐴)
45 ordn2lp 6338 . . 3 (Ord βˆͺ 𝐴 β†’ Β¬ (βˆͺ 𝐴 ∈ (harβ€˜βˆͺ 𝐴) ∧ (harβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴))
469, 44, 453syl 18 . 2 (𝐴 ∈ V β†’ Β¬ (βˆͺ 𝐴 ∈ (harβ€˜βˆͺ 𝐴) ∧ (harβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴))
4743, 46pm2.65i 193 1 Β¬ 𝐴 ∈ V
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  Vcvv 3444   βŠ† wss 3911  βˆͺ cuni 4866   class class class wbr 5106  Ord word 6317  Oncon0 6318  β€˜cfv 6497   β‰Ό cdom 8884   β‰Ί csdm 8885  harchar 9497  cardccrd 9876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-oi 9451  df-har 9498  df-card 9880
This theorem is referenced by:  cardprc  9921
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