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Mirrors > Home > NFE Home > Th. List > nchoicelem7 | GIF version |
Description: Lemma for nchoice 6306. Calculate the cardinality of a special set generator. Theorem 6.7 of [Specker] p. 974. (Contributed by SF, 13-Mar-2015.) |
Ref | Expression |
---|---|
nchoicelem7 | ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → Nc ( Spac ‘M) = ( Nc ( Spac ‘(2c ↑c M)) +c 1c)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nchoicelem6 6292 | . . 3 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ( Spac ‘M) = ({M} ∪ ( Spac ‘(2c ↑c M)))) | |
2 | 1 | nceqd 6110 | . 2 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → Nc ( Spac ‘M) = Nc ({M} ∪ ( Spac ‘(2c ↑c M)))) |
3 | incom 3448 | . . . . . 6 ⊢ ({M} ∩ ( Spac ‘(2c ↑c M))) = (( Spac ‘(2c ↑c M)) ∩ {M}) | |
4 | nchoicelem5 6291 | . . . . . . 7 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ¬ M ∈ ( Spac ‘(2c ↑c M))) | |
5 | disjsn 3786 | . . . . . . 7 ⊢ ((( Spac ‘(2c ↑c M)) ∩ {M}) = ∅ ↔ ¬ M ∈ ( Spac ‘(2c ↑c M))) | |
6 | 4, 5 | sylibr 203 | . . . . . 6 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (( Spac ‘(2c ↑c M)) ∩ {M}) = ∅) |
7 | 3, 6 | syl5eq 2397 | . . . . 5 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → ({M} ∩ ( Spac ‘(2c ↑c M))) = ∅) |
8 | snex 4111 | . . . . . 6 ⊢ {M} ∈ V | |
9 | fvex 5339 | . . . . . 6 ⊢ ( Spac ‘(2c ↑c M)) ∈ V | |
10 | 8, 9 | ncdisjun 6136 | . . . . 5 ⊢ (({M} ∩ ( Spac ‘(2c ↑c M))) = ∅ → Nc ({M} ∪ ( Spac ‘(2c ↑c M))) = ( Nc {M} +c Nc ( Spac ‘(2c ↑c M)))) |
11 | 7, 10 | syl 15 | . . . 4 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → Nc ({M} ∪ ( Spac ‘(2c ↑c M))) = ( Nc {M} +c Nc ( Spac ‘(2c ↑c M)))) |
12 | df1c3g 6141 | . . . . . 6 ⊢ (M ∈ NC → 1c = Nc {M}) | |
13 | 12 | addceq1d 4389 | . . . . 5 ⊢ (M ∈ NC → (1c +c Nc ( Spac ‘(2c ↑c M))) = ( Nc {M} +c Nc ( Spac ‘(2c ↑c M)))) |
14 | 13 | adantr 451 | . . . 4 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → (1c +c Nc ( Spac ‘(2c ↑c M))) = ( Nc {M} +c Nc ( Spac ‘(2c ↑c M)))) |
15 | 11, 14 | eqtr4d 2388 | . . 3 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → Nc ({M} ∪ ( Spac ‘(2c ↑c M))) = (1c +c Nc ( Spac ‘(2c ↑c M)))) |
16 | addccom 4406 | . . 3 ⊢ (1c +c Nc ( Spac ‘(2c ↑c M))) = ( Nc ( Spac ‘(2c ↑c M)) +c 1c) | |
17 | 15, 16 | syl6eq 2401 | . 2 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → Nc ({M} ∪ ( Spac ‘(2c ↑c M))) = ( Nc ( Spac ‘(2c ↑c M)) +c 1c)) |
18 | 2, 17 | eqtrd 2385 | 1 ⊢ ((M ∈ NC ∧ (M ↑c 0c) ∈ NC ) → Nc ( Spac ‘M) = ( Nc ( Spac ‘(2c ↑c M)) +c 1c)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∪ cun 3207 ∩ cin 3208 ∅c0 3550 {csn 3737 1cc1c 4134 0cc0c 4374 +c cplc 4375 ‘cfv 4781 (class class class)co 5525 NC cncs 6088 Nc cnc 6091 2cc2c 6094 ↑c cce 6096 Spac cspac 6271 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-fv 4795 df-2nd 4797 df-ov 5526 df-oprab 5528 df-mpt 5652 df-mpt2 5654 df-txp 5736 df-fix 5740 df-compose 5748 df-ins2 5750 df-ins3 5752 df-image 5754 df-ins4 5756 df-si3 5758 df-funs 5760 df-fns 5762 df-pw1fn 5766 df-fullfun 5768 df-clos1 5873 df-trans 5899 df-sym 5908 df-er 5909 df-ec 5947 df-qs 5951 df-map 6001 df-en 6029 df-ncs 6098 df-lec 6099 df-ltc 6100 df-nc 6101 df-2c 6104 df-ce 6106 df-spac 6272 |
This theorem is referenced by: nchoicelem9 6295 nchoicelem12 6298 nchoicelem17 6303 |
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