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Mirrors > Home > NFE Home > Th. List > ceclb | GIF version |
Description: Biconditional closure law for cardinal exponentiation. Theorem XI.2.48 of [Rosser] p. 384. (Contributed by SF, 9-Mar-2015.) |
Ref | Expression |
---|---|
ceclb | ⊢ ((M ∈ NC ∧ N ∈ NC ) → (((M ↑c 0c) ≠ ∅ ∧ (N ↑c 0c) ≠ ∅) ↔ (M ↑c N) ∈ NC )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ce0nnul 6177 | . . . . 5 ⊢ (M ∈ NC → ((M ↑c 0c) ≠ ∅ ↔ ∃a℘1a ∈ M)) | |
2 | ce0nnul 6177 | . . . . 5 ⊢ (N ∈ NC → ((N ↑c 0c) ≠ ∅ ↔ ∃b℘1b ∈ N)) | |
3 | 1, 2 | bi2anan9 843 | . . . 4 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (((M ↑c 0c) ≠ ∅ ∧ (N ↑c 0c) ≠ ∅) ↔ (∃a℘1a ∈ M ∧ ∃b℘1b ∈ N))) |
4 | eeanv 1913 | . . . 4 ⊢ (∃a∃b(℘1a ∈ M ∧ ℘1b ∈ N) ↔ (∃a℘1a ∈ M ∧ ∃b℘1b ∈ N)) | |
5 | 3, 4 | syl6bbr 254 | . . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (((M ↑c 0c) ≠ ∅ ∧ (N ↑c 0c) ≠ ∅) ↔ ∃a∃b(℘1a ∈ M ∧ ℘1b ∈ N))) |
6 | ncseqnc 6128 | . . . . . 6 ⊢ (M ∈ NC → (M = Nc ℘1a ↔ ℘1a ∈ M)) | |
7 | ncseqnc 6128 | . . . . . 6 ⊢ (N ∈ NC → (N = Nc ℘1b ↔ ℘1b ∈ N)) | |
8 | 6, 7 | bi2anan9 843 | . . . . 5 ⊢ ((M ∈ NC ∧ N ∈ NC ) → ((M = Nc ℘1a ∧ N = Nc ℘1b) ↔ (℘1a ∈ M ∧ ℘1b ∈ N))) |
9 | oveq12 5532 | . . . . . 6 ⊢ ((M = Nc ℘1a ∧ N = Nc ℘1b) → (M ↑c N) = ( Nc ℘1a ↑c Nc ℘1b)) | |
10 | vex 2862 | . . . . . . . 8 ⊢ a ∈ V | |
11 | vex 2862 | . . . . . . . 8 ⊢ b ∈ V | |
12 | 10, 11 | cenc 6181 | . . . . . . 7 ⊢ ( Nc ℘1a ↑c Nc ℘1b) = Nc (a ↑m b) |
13 | ovex 5551 | . . . . . . . 8 ⊢ (a ↑m b) ∈ V | |
14 | 13 | ncelncsi 6121 | . . . . . . 7 ⊢ Nc (a ↑m b) ∈ NC |
15 | 12, 14 | eqeltri 2423 | . . . . . 6 ⊢ ( Nc ℘1a ↑c Nc ℘1b) ∈ NC |
16 | 9, 15 | syl6eqel 2441 | . . . . 5 ⊢ ((M = Nc ℘1a ∧ N = Nc ℘1b) → (M ↑c N) ∈ NC ) |
17 | 8, 16 | syl6bir 220 | . . . 4 ⊢ ((M ∈ NC ∧ N ∈ NC ) → ((℘1a ∈ M ∧ ℘1b ∈ N) → (M ↑c N) ∈ NC )) |
18 | 17 | exlimdvv 1637 | . . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (∃a∃b(℘1a ∈ M ∧ ℘1b ∈ N) → (M ↑c N) ∈ NC )) |
19 | 5, 18 | sylbid 206 | . 2 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (((M ↑c 0c) ≠ ∅ ∧ (N ↑c 0c) ≠ ∅) → (M ↑c N) ∈ NC )) |
20 | nulnnc 6118 | . . . . 5 ⊢ ¬ ∅ ∈ NC | |
21 | eleq1 2413 | . . . . 5 ⊢ ((M ↑c N) = ∅ → ((M ↑c N) ∈ NC ↔ ∅ ∈ NC )) | |
22 | 20, 21 | mtbiri 294 | . . . 4 ⊢ ((M ↑c N) = ∅ → ¬ (M ↑c N) ∈ NC ) |
23 | 22 | necon2ai 2561 | . . 3 ⊢ ((M ↑c N) ∈ NC → (M ↑c N) ≠ ∅) |
24 | ce0nnulb 6182 | . . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (((M ↑c 0c) ≠ ∅ ∧ (N ↑c 0c) ≠ ∅) ↔ (M ↑c N) ≠ ∅)) | |
25 | 23, 24 | syl5ibr 212 | . 2 ⊢ ((M ∈ NC ∧ N ∈ NC ) → ((M ↑c N) ∈ NC → ((M ↑c 0c) ≠ ∅ ∧ (N ↑c 0c) ≠ ∅))) |
26 | 19, 25 | impbid 183 | 1 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (((M ↑c 0c) ≠ ∅ ∧ (N ↑c 0c) ≠ ∅) ↔ (M ↑c N) ∈ NC )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∅c0 3550 ℘1cpw1 4135 0cc0c 4374 (class class class)co 5525 ↑m cmap 5999 NC cncs 6088 Nc cnc 6091 ↑c cce 6096 |
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-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-trans 5899 df-sym 5908 df-er 5909 df-ec 5947 df-qs 5951 df-map 6001 df-en 6029 df-ncs 6098 df-nc 6101 df-ce 6106 |
This theorem is referenced by: ce0nulnc 6184 cecl 6186 ceclr 6187 fce 6188 |
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