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Mirrors > Home > NFE Home > Th. List > nchoicelem4 | GIF version |
Description: Lemma for nchoice 6308. The initial value of Spac is a minimum value. Theorem 6.4 of [Specker] p. 973. (Contributed by SF, 13-Mar-2015.) |
Ref | Expression |
---|---|
nchoicelem4 | ⊢ ((M ∈ NC ∧ N ∈ ( Spac ‘M)) → M ≤c N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasn 5018 | . . 3 ⊢ ( ≤c “ {M}) = {p ∣ M ≤c p} | |
2 | lecex 6115 | . . . 4 ⊢ ≤c ∈ V | |
3 | snex 4111 | . . . 4 ⊢ {M} ∈ V | |
4 | 2, 3 | imaex 4747 | . . 3 ⊢ ( ≤c “ {M}) ∈ V |
5 | 1, 4 | eqeltrri 2424 | . 2 ⊢ {p ∣ M ≤c p} ∈ V |
6 | breq2 4643 | . 2 ⊢ (p = M → (M ≤c p ↔ M ≤c M)) | |
7 | breq2 4643 | . 2 ⊢ (p = n → (M ≤c p ↔ M ≤c n)) | |
8 | breq2 4643 | . 2 ⊢ (p = (2c ↑c n) → (M ≤c p ↔ M ≤c (2c ↑c n))) | |
9 | breq2 4643 | . 2 ⊢ (p = N → (M ≤c p ↔ M ≤c N)) | |
10 | nclecid 6197 | . 2 ⊢ (M ∈ NC → M ≤c M) | |
11 | spacssnc 6284 | . . . . . . 7 ⊢ (M ∈ NC → ( Spac ‘M) ⊆ NC ) | |
12 | 11 | sselda 3273 | . . . . . 6 ⊢ ((M ∈ NC ∧ n ∈ ( Spac ‘M)) → n ∈ NC ) |
13 | ce2lt 6220 | . . . . . 6 ⊢ ((n ∈ NC ∧ (n ↑c 0c) ∈ NC ) → n <c (2c ↑c n)) | |
14 | 12, 13 | sylan 457 | . . . . 5 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ (n ↑c 0c) ∈ NC ) → n <c (2c ↑c n)) |
15 | brltc 6114 | . . . . . 6 ⊢ (n <c (2c ↑c n) ↔ (n ≤c (2c ↑c n) ∧ n ≠ (2c ↑c n))) | |
16 | 15 | simplbi 446 | . . . . 5 ⊢ (n <c (2c ↑c n) → n ≤c (2c ↑c n)) |
17 | 14, 16 | syl 15 | . . . 4 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ (n ↑c 0c) ∈ NC ) → n ≤c (2c ↑c n)) |
18 | simpll 730 | . . . . 5 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ (n ↑c 0c) ∈ NC ) → M ∈ NC ) | |
19 | 12 | adantr 451 | . . . . 5 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ (n ↑c 0c) ∈ NC ) → n ∈ NC ) |
20 | 2nnc 6167 | . . . . . . 7 ⊢ 2c ∈ Nn | |
21 | ceclnn1 6189 | . . . . . . 7 ⊢ ((2c ∈ Nn ∧ n ∈ NC ∧ (n ↑c 0c) ∈ NC ) → (2c ↑c n) ∈ NC ) | |
22 | 20, 21 | mp3an1 1264 | . . . . . 6 ⊢ ((n ∈ NC ∧ (n ↑c 0c) ∈ NC ) → (2c ↑c n) ∈ NC ) |
23 | 12, 22 | sylan 457 | . . . . 5 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ (n ↑c 0c) ∈ NC ) → (2c ↑c n) ∈ NC ) |
24 | lectr 6211 | . . . . 5 ⊢ ((M ∈ NC ∧ n ∈ NC ∧ (2c ↑c n) ∈ NC ) → ((M ≤c n ∧ n ≤c (2c ↑c n)) → M ≤c (2c ↑c n))) | |
25 | 18, 19, 23, 24 | syl3anc 1182 | . . . 4 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ (n ↑c 0c) ∈ NC ) → ((M ≤c n ∧ n ≤c (2c ↑c n)) → M ≤c (2c ↑c n))) |
26 | 17, 25 | mpan2d 655 | . . 3 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ (n ↑c 0c) ∈ NC ) → (M ≤c n → M ≤c (2c ↑c n))) |
27 | 26 | impr 602 | . 2 ⊢ (((M ∈ NC ∧ n ∈ ( Spac ‘M)) ∧ ((n ↑c 0c) ∈ NC ∧ M ≤c n)) → M ≤c (2c ↑c n)) |
28 | 5, 6, 7, 8, 9, 10, 27 | spacis 6288 | 1 ⊢ ((M ∈ NC ∧ N ∈ ( Spac ‘M)) → M ≤c N) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∈ wcel 1710 {cab 2339 ≠ wne 2516 Vcvv 2859 {csn 3737 Nn cnnc 4373 0cc0c 4374 class class class wbr 4639 “ cima 4722 ‘cfv 4781 (class class class)co 5525 NC cncs 6088 ≤c clec 6089 <c cltc 6090 2cc2c 6094 ↑c cce 6096 Spac cspac 6273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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 6274 |
This theorem is referenced by: nchoicelem5 6293 |
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