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Theorem cardprclem 9976
Description: Lemma for cardprc 9977. (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 2819 . . . . . . . 8 (π‘₯ ∈ 𝐴 ↔ π‘₯ ∈ {π‘₯ ∣ (cardβ€˜π‘₯) = π‘₯})
3 abid 2707 . . . . . . . 8 (π‘₯ ∈ {π‘₯ ∣ (cardβ€˜π‘₯) = π‘₯} ↔ (cardβ€˜π‘₯) = π‘₯)
4 iscard 9972 . . . . . . . 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 497 . . . . . 6 (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ On)
76ssriv 3981 . . . . 5 𝐴 βŠ† On
8 ssonuni 7764 . . . . 5 (𝐴 ∈ V β†’ (𝐴 βŠ† On β†’ βˆͺ 𝐴 ∈ On))
97, 8mpi 20 . . . 4 (𝐴 ∈ V β†’ βˆͺ 𝐴 ∈ On)
10 domrefg 8985 . . . . 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 9558 . . . 4 (βˆͺ 𝐴 ∈ (harβ€˜βˆͺ 𝐴) ↔ (βˆͺ 𝐴 ∈ On ∧ βˆͺ 𝐴 β‰Ό βˆͺ 𝐴))
139, 11, 12sylanbrc 582 . . 3 (𝐴 ∈ V β†’ βˆͺ 𝐴 ∈ (harβ€˜βˆͺ 𝐴))
147sseli 3973 . . . . . . . 8 (𝑧 ∈ 𝐴 β†’ 𝑧 ∈ On)
15 domrefg 8985 . . . . . . . . . 10 (𝑧 ∈ On β†’ 𝑧 β‰Ό 𝑧)
1615ancli 548 . . . . . . . . 9 (𝑧 ∈ On β†’ (𝑧 ∈ On ∧ 𝑧 β‰Ό 𝑧))
17 elharval 9558 . . . . . . . . 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 9975 . . . . . . . 8 (cardβ€˜(harβ€˜π‘§)) = (harβ€˜π‘§)
21 fvex 6898 . . . . . . . . 9 (harβ€˜π‘§) ∈ V
22 fveq2 6885 . . . . . . . . . 10 (π‘₯ = (harβ€˜π‘§) β†’ (cardβ€˜π‘₯) = (cardβ€˜(harβ€˜π‘§)))
23 id 22 . . . . . . . . . 10 (π‘₯ = (harβ€˜π‘§) β†’ π‘₯ = (harβ€˜π‘§))
2422, 23eqeq12d 2742 . . . . . . . . 9 (π‘₯ = (harβ€˜π‘§) β†’ ((cardβ€˜π‘₯) = π‘₯ ↔ (cardβ€˜(harβ€˜π‘§)) = (harβ€˜π‘§)))
2521, 24, 1elab2 3667 . . . . . . . 8 ((harβ€˜π‘§) ∈ 𝐴 ↔ (cardβ€˜(harβ€˜π‘§)) = (harβ€˜π‘§))
2620, 25mpbir 230 . . . . . . 7 (harβ€˜π‘§) ∈ 𝐴
27 eleq2 2816 . . . . . . . . 9 (𝑀 = (harβ€˜π‘§) β†’ (𝑧 ∈ 𝑀 ↔ 𝑧 ∈ (harβ€˜π‘§)))
28 eleq1 2815 . . . . . . . . 9 (𝑀 = (harβ€˜π‘§) β†’ (𝑀 ∈ 𝐴 ↔ (harβ€˜π‘§) ∈ 𝐴))
2927, 28anbi12d 630 . . . . . . . 8 (𝑀 = (harβ€˜π‘§) β†’ ((𝑧 ∈ 𝑀 ∧ 𝑀 ∈ 𝐴) ↔ (𝑧 ∈ (harβ€˜π‘§) ∧ (harβ€˜π‘§) ∈ 𝐴)))
3021, 29spcev 3590 . . . . . . 7 ((𝑧 ∈ (harβ€˜π‘§) ∧ (harβ€˜π‘§) ∈ 𝐴) β†’ βˆƒπ‘€(𝑧 ∈ 𝑀 ∧ 𝑀 ∈ 𝐴))
3119, 26, 30sylancl 585 . . . . . 6 (𝑧 ∈ 𝐴 β†’ βˆƒπ‘€(𝑧 ∈ 𝑀 ∧ 𝑀 ∈ 𝐴))
32 eluni 4905 . . . . . 6 (𝑧 ∈ βˆͺ 𝐴 ↔ βˆƒπ‘€(𝑧 ∈ 𝑀 ∧ 𝑀 ∈ 𝐴))
3331, 32sylibr 233 . . . . 5 (𝑧 ∈ 𝐴 β†’ 𝑧 ∈ βˆͺ 𝐴)
3433ssriv 3981 . . . 4 𝐴 βŠ† βˆͺ 𝐴
35 harcard 9975 . . . . 5 (cardβ€˜(harβ€˜βˆͺ 𝐴)) = (harβ€˜βˆͺ 𝐴)
36 fvex 6898 . . . . . 6 (harβ€˜βˆͺ 𝐴) ∈ V
37 fveq2 6885 . . . . . . 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 2742 . . . . . 6 (π‘₯ = (harβ€˜βˆͺ 𝐴) β†’ ((cardβ€˜π‘₯) = π‘₯ ↔ (cardβ€˜(harβ€˜βˆͺ 𝐴)) = (harβ€˜βˆͺ 𝐴)))
4036, 39, 1elab2 3667 . . . . 5 ((harβ€˜βˆͺ 𝐴) ∈ 𝐴 ↔ (cardβ€˜(harβ€˜βˆͺ 𝐴)) = (harβ€˜βˆͺ 𝐴))
4135, 40mpbir 230 . . . 4 (harβ€˜βˆͺ 𝐴) ∈ 𝐴
4234, 41sselii 3974 . . 3 (harβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴
4313, 42jctir 520 . 2 (𝐴 ∈ V β†’ (βˆͺ 𝐴 ∈ (harβ€˜βˆͺ 𝐴) ∧ (harβ€˜βˆͺ 𝐴) ∈ βˆͺ 𝐴))
44 eloni 6368 . . 3 (βˆͺ 𝐴 ∈ On β†’ Ord βˆͺ 𝐴)
45 ordn2lp 6378 . . 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 395   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  {cab 2703  βˆ€wral 3055  Vcvv 3468   βŠ† wss 3943  βˆͺ cuni 4902   class class class wbr 5141  Ord word 6357  Oncon0 6358  β€˜cfv 6537   β‰Ό cdom 8939   β‰Ί csdm 8940  harchar 9553  cardccrd 9932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-oi 9507  df-har 9554  df-card 9936
This theorem is referenced by:  cardprc  9977
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