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Theorem nchoicelem9 6297
 Description: Lemma for nchoice 6308. Calculate the cardinality of the special set generator when near the end of raisability. Theorem 6.8 of [Specker] p. 974. (Contributed by SF, 18-Mar-2015.)
Assertion
Ref Expression
nchoicelem9 (( ≤c We NC M NC ¬ (Mc 0c) NC ) → ( Nc ( SpacTc M) = 2c Nc ( SpacTc M) = 3c))

Proof of Theorem nchoicelem9
StepHypRef Expression
1 brltc 6114 . . . . 5 ( Nc 1c <c M ↔ ( Nc 1cc M Nc 1cM))
21simplbi 446 . . . 4 ( Nc 1c <c MNc 1cc M)
3 1cex 4142 . . . . . . . 8 1c V
43ncelncsi 6121 . . . . . . 7 Nc 1c NC
5 tlecg 6230 . . . . . . 7 (( Nc 1c NC M NC ) → ( Nc 1cc MTc Nc 1cc Tc M))
64, 5mpan 651 . . . . . 6 (M NC → ( Nc 1cc MTc Nc 1cc Tc M))
76adantl 452 . . . . 5 (( ≤c We NC M NC ) → ( Nc 1cc MTc Nc 1cc Tc M))
8 tcnc1c 6227 . . . . . . 7 Tc Nc 1c = Nc 11c
98breq1i 4646 . . . . . 6 ( Tc Nc 1cc Tc MNc 11cc Tc M)
10 tccl 6160 . . . . . . . . 9 (M NCTc M NC )
11 te0c 6237 . . . . . . . . 9 (M NC → ( Tc Mc 0c) NC )
123pw1ex 4303 . . . . . . . . . . 11 11c V
1312ncelncsi 6121 . . . . . . . . . 10 Nc 11c NC
14 ce2le 6233 . . . . . . . . . . 11 ((( Nc 11c NC Tc M NC ( Tc Mc 0c) NC ) Nc 11cc Tc M) → (2cc Nc 11c) ≤c (2cc Tc M))
1514ex 423 . . . . . . . . . 10 (( Nc 11c NC Tc M NC ( Tc Mc 0c) NC ) → ( Nc 11cc Tc M → (2cc Nc 11c) ≤c (2cc Tc M)))
1613, 15mp3an1 1264 . . . . . . . . 9 (( Tc M NC ( Tc Mc 0c) NC ) → ( Nc 11cc Tc M → (2cc Nc 11c) ≤c (2cc Tc M)))
1710, 11, 16syl2anc 642 . . . . . . . 8 (M NC → ( Nc 11cc Tc M → (2cc Nc 11c) ≤c (2cc Tc M)))
1817adantl 452 . . . . . . 7 (( ≤c We NC M NC ) → ( Nc 11cc Tc M → (2cc Nc 11c) ≤c (2cc Tc M)))
19 ce2ncpw11c 6194 . . . . . . . . 9 (2cc Nc 11c) = Nc 1c
2019breq1i 4646 . . . . . . . 8 ((2cc Nc 11c) ≤c (2cc Tc M) ↔ Nc 1cc (2cc Tc M))
21 orc 374 . . . . . . . . . 10 ( Nc 1cc (2cc Tc M) → ( Nc 1cc (2cc Tc M) Nc 1c = (2cc Tc M)))
22 brltc 6114 . . . . . . . . . . . 12 ( Nc 1c <c (2cc Tc M) ↔ ( Nc 1cc (2cc Tc M) Nc 1c ≠ (2cc Tc M)))
2322orbi1i 506 . . . . . . . . . . 11 (( Nc 1c <c (2cc Tc M) Nc 1c = (2cc Tc M)) ↔ (( Nc 1cc (2cc Tc M) Nc 1c ≠ (2cc Tc M)) Nc 1c = (2cc Tc M)))
24 pm2.1 406 . . . . . . . . . . . . 13 Nc 1c = (2cc Tc M) Nc 1c = (2cc Tc M))
25 df-ne 2518 . . . . . . . . . . . . . 14 ( Nc 1c ≠ (2cc Tc M) ↔ ¬ Nc 1c = (2cc Tc M))
2625orbi1i 506 . . . . . . . . . . . . 13 (( Nc 1c ≠ (2cc Tc M) Nc 1c = (2cc Tc M)) ↔ (¬ Nc 1c = (2cc Tc M) Nc 1c = (2cc Tc M)))
2724, 26mpbir 200 . . . . . . . . . . . 12 ( Nc 1c ≠ (2cc Tc M) Nc 1c = (2cc Tc M))
28 ordir 835 . . . . . . . . . . . 12 ((( Nc 1cc (2cc Tc M) Nc 1c ≠ (2cc Tc M)) Nc 1c = (2cc Tc M)) ↔ (( Nc 1cc (2cc Tc M) Nc 1c = (2cc Tc M)) ( Nc 1c ≠ (2cc Tc M) Nc 1c = (2cc Tc M))))
2927, 28mpbiran2 885 . . . . . . . . . . 11 ((( Nc 1cc (2cc Tc M) Nc 1c ≠ (2cc Tc M)) Nc 1c = (2cc Tc M)) ↔ ( Nc 1cc (2cc Tc M) Nc 1c = (2cc Tc M)))
3023, 29bitri 240 . . . . . . . . . 10 (( Nc 1c <c (2cc Tc M) Nc 1c = (2cc Tc M)) ↔ ( Nc 1cc (2cc Tc M) Nc 1c = (2cc Tc M)))
3121, 30sylibr 203 . . . . . . . . 9 ( Nc 1cc (2cc Tc M) → ( Nc 1c <c (2cc Tc M) Nc 1c = (2cc Tc M)))
32 ce2t 6235 . . . . . . . . . . . 12 (M NC → (2cc Tc M) NC )
33 nchoicelem8 6296 . . . . . . . . . . . 12 (( ≤c We NC (2cc Tc M) NC ) → (¬ ((2cc Tc M) ↑c 0c) NCNc 1c <c (2cc Tc M)))
3432, 33sylan2 460 . . . . . . . . . . 11 (( ≤c We NC M NC ) → (¬ ((2cc Tc M) ↑c 0c) NCNc 1c <c (2cc Tc M)))
35 nchoicelem3 6291 . . . . . . . . . . . . . . . 16 (((2cc Tc M) NC ¬ ((2cc Tc M) ↑c 0c) NC ) → ( Spac ‘(2cc Tc M)) = {(2cc Tc M)})
3635nceqd 6110 . . . . . . . . . . . . . . 15 (((2cc Tc M) NC ¬ ((2cc Tc M) ↑c 0c) NC ) → Nc ( Spac ‘(2cc Tc M)) = Nc {(2cc Tc M)})
37 ovex 5551 . . . . . . . . . . . . . . . 16 (2cc Tc M) V
3837df1c3 6140 . . . . . . . . . . . . . . 15 1c = Nc {(2cc Tc M)}
3936, 38syl6eqr 2403 . . . . . . . . . . . . . 14 (((2cc Tc M) NC ¬ ((2cc Tc M) ↑c 0c) NC ) → Nc ( Spac ‘(2cc Tc M)) = 1c)
4039ex 423 . . . . . . . . . . . . 13 ((2cc Tc M) NC → (¬ ((2cc Tc M) ↑c 0c) NCNc ( Spac ‘(2cc Tc M)) = 1c))
4132, 40syl 15 . . . . . . . . . . . 12 (M NC → (¬ ((2cc Tc M) ↑c 0c) NCNc ( Spac ‘(2cc Tc M)) = 1c))
4241adantl 452 . . . . . . . . . . 11 (( ≤c We NC M NC ) → (¬ ((2cc Tc M) ↑c 0c) NCNc ( Spac ‘(2cc Tc M)) = 1c))
4334, 42sylbird 226 . . . . . . . . . 10 (( ≤c We NC M NC ) → ( Nc 1c <c (2cc Tc M) → Nc ( Spac ‘(2cc Tc M)) = 1c))
44 nclecid 6197 . . . . . . . . . . . . . . . . . . 19 ( Nc 1c NCNc 1cc Nc 1c)
454, 44ax-mp 8 . . . . . . . . . . . . . . . . . 18 Nc 1cc Nc 1c
46 ce0lenc1 6239 . . . . . . . . . . . . . . . . . . 19 ( Nc 1c NC → (( Nc 1cc 0c) NCNc 1cc Nc 1c))
474, 46ax-mp 8 . . . . . . . . . . . . . . . . . 18 (( Nc 1cc 0c) NCNc 1cc Nc 1c)
4845, 47mpbir 200 . . . . . . . . . . . . . . . . 17 ( Nc 1cc 0c) NC
49 ce2lt 6220 . . . . . . . . . . . . . . . . 17 (( Nc 1c NC ( Nc 1cc 0c) NC ) → Nc 1c <c (2cc Nc 1c))
504, 48, 49mp2an 653 . . . . . . . . . . . . . . . 16 Nc 1c <c (2cc Nc 1c)
51 2nnc 6167 . . . . . . . . . . . . . . . . . 18 2c Nn
52 ceclnn1 6189 . . . . . . . . . . . . . . . . . 18 ((2c Nn Nc 1c NC ( Nc 1cc 0c) NC ) → (2cc Nc 1c) NC )
5351, 4, 48, 52mp3an 1277 . . . . . . . . . . . . . . . . 17 (2cc Nc 1c) NC
54 nchoicelem8 6296 . . . . . . . . . . . . . . . . 17 (( ≤c We NC (2cc Nc 1c) NC ) → (¬ ((2cc Nc 1c) ↑c 0c) NCNc 1c <c (2cc Nc 1c)))
5553, 54mpan2 652 . . . . . . . . . . . . . . . 16 ( ≤c We NC → (¬ ((2cc Nc 1c) ↑c 0c) NCNc 1c <c (2cc Nc 1c)))
5650, 55mpbiri 224 . . . . . . . . . . . . . . 15 ( ≤c We NC → ¬ ((2cc Nc 1c) ↑c 0c) NC )
57 nchoicelem3 6291 . . . . . . . . . . . . . . . . . 18 (((2cc Nc 1c) NC ¬ ((2cc Nc 1c) ↑c 0c) NC ) → ( Spac ‘(2cc Nc 1c)) = {(2cc Nc 1c)})
5857nceqd 6110 . . . . . . . . . . . . . . . . 17 (((2cc Nc 1c) NC ¬ ((2cc Nc 1c) ↑c 0c) NC ) → Nc ( Spac ‘(2cc Nc 1c)) = Nc {(2cc Nc 1c)})
59 ovex 5551 . . . . . . . . . . . . . . . . . 18 (2cc Nc 1c) V
6059df1c3 6140 . . . . . . . . . . . . . . . . 17 1c = Nc {(2cc Nc 1c)}
6158, 60syl6eqr 2403 . . . . . . . . . . . . . . . 16 (((2cc Nc 1c) NC ¬ ((2cc Nc 1c) ↑c 0c) NC ) → Nc ( Spac ‘(2cc Nc 1c)) = 1c)
6253, 61mpan 651 . . . . . . . . . . . . . . 15 (¬ ((2cc Nc 1c) ↑c 0c) NCNc ( Spac ‘(2cc Nc 1c)) = 1c)
6356, 62syl 15 . . . . . . . . . . . . . 14 ( ≤c We NCNc ( Spac ‘(2cc Nc 1c)) = 1c)
6463addceq1d 4389 . . . . . . . . . . . . 13 ( ≤c We NC → ( Nc ( Spac ‘(2cc Nc 1c)) +c 1c) = (1c +c 1c))
65 nchoicelem7 6295 . . . . . . . . . . . . . 14 (( Nc 1c NC ( Nc 1cc 0c) NC ) → Nc ( SpacNc 1c) = ( Nc ( Spac ‘(2cc Nc 1c)) +c 1c))
664, 48, 65mp2an 653 . . . . . . . . . . . . 13 Nc ( SpacNc 1c) = ( Nc ( Spac ‘(2cc Nc 1c)) +c 1c)
67 1p1e2c 6155 . . . . . . . . . . . . . 14 (1c +c 1c) = 2c
6867eqcomi 2357 . . . . . . . . . . . . 13 2c = (1c +c 1c)
6964, 66, 683eqtr4g 2410 . . . . . . . . . . . 12 ( ≤c We NCNc ( SpacNc 1c) = 2c)
70 fveq2 5328 . . . . . . . . . . . . . 14 ( Nc 1c = (2cc Tc M) → ( SpacNc 1c) = ( Spac ‘(2cc Tc M)))
7170nceqd 6110 . . . . . . . . . . . . 13 ( Nc 1c = (2cc Tc M) → Nc ( SpacNc 1c) = Nc ( Spac ‘(2cc Tc M)))
7271eqeq1d 2361 . . . . . . . . . . . 12 ( Nc 1c = (2cc Tc M) → ( Nc ( SpacNc 1c) = 2cNc ( Spac ‘(2cc Tc M)) = 2c))
7369, 72syl5ibcom 211 . . . . . . . . . . 11 ( ≤c We NC → ( Nc 1c = (2cc Tc M) → Nc ( Spac ‘(2cc Tc M)) = 2c))
7473adantr 451 . . . . . . . . . 10 (( ≤c We NC M NC ) → ( Nc 1c = (2cc Tc M) → Nc ( Spac ‘(2cc Tc M)) = 2c))
7543, 74orim12d 811 . . . . . . . . 9 (( ≤c We NC M NC ) → (( Nc 1c <c (2cc Tc M) Nc 1c = (2cc Tc M)) → ( Nc ( Spac ‘(2cc Tc M)) = 1c Nc ( Spac ‘(2cc Tc M)) = 2c)))
7631, 75syl5 28 . . . . . . . 8 (( ≤c We NC M NC ) → ( Nc 1cc (2cc Tc M) → ( Nc ( Spac ‘(2cc Tc M)) = 1c Nc ( Spac ‘(2cc Tc M)) = 2c)))
7720, 76syl5bi 208 . . . . . . 7 (( ≤c We NC M NC ) → ((2cc Nc 11c) ≤c (2cc Tc M) → ( Nc ( Spac ‘(2cc Tc M)) = 1c Nc ( Spac ‘(2cc Tc M)) = 2c)))
7818, 77syld 40 . . . . . 6 (( ≤c We NC M NC ) → ( Nc 11cc Tc M → ( Nc ( Spac ‘(2cc Tc M)) = 1c Nc ( Spac ‘(2cc Tc M)) = 2c)))
799, 78syl5bi 208 . . . . 5 (( ≤c We NC M NC ) → ( Tc Nc 1cc Tc M → ( Nc ( Spac ‘(2cc Tc M)) = 1c Nc ( Spac ‘(2cc Tc M)) = 2c)))
807, 79sylbid 206 . . . 4 (( ≤c We NC M NC ) → ( Nc 1cc M → ( Nc ( Spac ‘(2cc Tc M)) = 1c Nc ( Spac ‘(2cc Tc M)) = 2c)))
81 addceq1 4383 . . . . 5 ( Nc ( Spac ‘(2cc Tc M)) = 1c → ( Nc ( Spac ‘(2cc Tc M)) +c 1c) = (1c +c 1c))
82 addceq1 4383 . . . . 5 ( Nc ( Spac ‘(2cc Tc M)) = 2c → ( Nc ( Spac ‘(2cc Tc M)) +c 1c) = (2c +c 1c))
8381, 82orim12i 502 . . . 4 (( Nc ( Spac ‘(2cc Tc M)) = 1c Nc ( Spac ‘(2cc Tc M)) = 2c) → (( Nc ( Spac ‘(2cc Tc M)) +c 1c) = (1c +c 1c) ( Nc ( Spac ‘(2cc Tc M)) +c 1c) = (2c +c 1c)))
842, 80, 83syl56 30 . . 3 (( ≤c We NC M NC ) → ( Nc 1c <c M → (( Nc ( Spac ‘(2cc Tc M)) +c 1c) = (1c +c 1c) ( Nc ( Spac ‘(2cc Tc M)) +c 1c) = (2c +c 1c))))
85 nchoicelem8 6296 . . 3 (( ≤c We NC M NC ) → (¬ (Mc 0c) NCNc 1c <c M))
86 nchoicelem7 6295 . . . . . . 7 (( Tc M NC ( Tc Mc 0c) NC ) → Nc ( SpacTc M) = ( Nc ( Spac ‘(2cc Tc M)) +c 1c))
8710, 11, 86syl2anc 642 . . . . . 6 (M NCNc ( SpacTc M) = ( Nc ( Spac ‘(2cc Tc M)) +c 1c))
8868a1i 10 . . . . . 6 (M NC → 2c = (1c +c 1c))
8987, 88eqeq12d 2367 . . . . 5 (M NC → ( Nc ( SpacTc M) = 2c ↔ ( Nc ( Spac ‘(2cc Tc M)) +c 1c) = (1c +c 1c)))
90 2p1e3c 6156 . . . . . . . 8 (2c +c 1c) = 3c
9190eqcomi 2357 . . . . . . 7 3c = (2c +c 1c)
9291a1i 10 . . . . . 6 (M NC → 3c = (2c +c 1c))
9387, 92eqeq12d 2367 . . . . 5 (M NC → ( Nc ( SpacTc M) = 3c ↔ ( Nc ( Spac ‘(2cc Tc M)) +c 1c) = (2c +c 1c)))
9489, 93orbi12d 690 . . . 4 (M NC → (( Nc ( SpacTc M) = 2c Nc ( SpacTc M) = 3c) ↔ (( Nc ( Spac ‘(2cc Tc M)) +c 1c) = (1c +c 1c) ( Nc ( Spac ‘(2cc Tc M)) +c 1c) = (2c +c 1c))))
9594adantl 452 . . 3 (( ≤c We NC M NC ) → (( Nc ( SpacTc M) = 2c Nc ( SpacTc M) = 3c) ↔ (( Nc ( Spac ‘(2cc Tc M)) +c 1c) = (1c +c 1c) ( Nc ( Spac ‘(2cc Tc M)) +c 1c) = (2c +c 1c))))
9684, 85, 953imtr4d 259 . 2 (( ≤c We NC M NC ) → (¬ (Mc 0c) NC → ( Nc ( SpacTc M) = 2c Nc ( SpacTc M) = 3c)))
97963impia 1148 1 (( ≤c We NC M NC ¬ (Mc 0c) NC ) → ( Nc ( SpacTc M) = 2c Nc ( SpacTc M) = 3c))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  {csn 3737  1cc1c 4134  ℘1cpw1 4135   Nn cnnc 4373  0cc0c 4374   +c cplc 4375   class class class wbr 4639   ‘cfv 4781  (class class class)co 5525   We cwe 5895   NC cncs 6088   ≤c clec 6089
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