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Mirrors > Home > NFE Home > Th. List > ce0nnul | GIF version |
Description: A condition for cardinal exponentiation being non-empty. Theorem XI.2.42 of [Rosser] p. 382. (Contributed by SF, 6-Mar-2015.) |
Ref | Expression |
---|---|
ce0nnul | ⊢ (M ∈ NC → ((M ↑c 0c) ≠ ∅ ↔ ∃a℘1a ∈ M)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cnc 6138 | . . . 4 ⊢ 0c ∈ NC | |
2 | elce 6175 | . . . 4 ⊢ ((M ∈ NC ∧ 0c ∈ NC ) → (g ∈ (M ↑c 0c) ↔ ∃a∃b(℘1a ∈ M ∧ ℘1b ∈ 0c ∧ g ≈ (a ↑m b)))) | |
3 | 1, 2 | mpan2 652 | . . 3 ⊢ (M ∈ NC → (g ∈ (M ↑c 0c) ↔ ∃a∃b(℘1a ∈ M ∧ ℘1b ∈ 0c ∧ g ≈ (a ↑m b)))) |
4 | 3 | exbidv 1626 | . 2 ⊢ (M ∈ NC → (∃g g ∈ (M ↑c 0c) ↔ ∃g∃a∃b(℘1a ∈ M ∧ ℘1b ∈ 0c ∧ g ≈ (a ↑m b)))) |
5 | n0 3559 | . 2 ⊢ ((M ↑c 0c) ≠ ∅ ↔ ∃g g ∈ (M ↑c 0c)) | |
6 | 19.42vv 1907 | . . . . 5 ⊢ (∃g∃b(℘1a ∈ M ∧ (℘1b ∈ 0c ∧ g ≈ (a ↑m b))) ↔ (℘1a ∈ M ∧ ∃g∃b(℘1b ∈ 0c ∧ g ≈ (a ↑m b)))) | |
7 | 3anass 938 | . . . . . 6 ⊢ ((℘1a ∈ M ∧ ℘1b ∈ 0c ∧ g ≈ (a ↑m b)) ↔ (℘1a ∈ M ∧ (℘1b ∈ 0c ∧ g ≈ (a ↑m b)))) | |
8 | 7 | 2exbii 1583 | . . . . 5 ⊢ (∃g∃b(℘1a ∈ M ∧ ℘1b ∈ 0c ∧ g ≈ (a ↑m b)) ↔ ∃g∃b(℘1a ∈ M ∧ (℘1b ∈ 0c ∧ g ≈ (a ↑m b)))) |
9 | nulel0c 4422 | . . . . . . 7 ⊢ ∅ ∈ 0c | |
10 | ovex 5551 | . . . . . . . 8 ⊢ (a ↑m ∅) ∈ V | |
11 | 10 | enrflx 6035 | . . . . . . 7 ⊢ (a ↑m ∅) ≈ (a ↑m ∅) |
12 | 0ex 4110 | . . . . . . . 8 ⊢ ∅ ∈ V | |
13 | pw1eq 4143 | . . . . . . . . . . . 12 ⊢ (b = ∅ → ℘1b = ℘1∅) | |
14 | pw10 4161 | . . . . . . . . . . . 12 ⊢ ℘1∅ = ∅ | |
15 | 13, 14 | syl6eq 2401 | . . . . . . . . . . 11 ⊢ (b = ∅ → ℘1b = ∅) |
16 | 15 | eleq1d 2419 | . . . . . . . . . 10 ⊢ (b = ∅ → (℘1b ∈ 0c ↔ ∅ ∈ 0c)) |
17 | 16 | adantl 452 | . . . . . . . . 9 ⊢ ((g = (a ↑m ∅) ∧ b = ∅) → (℘1b ∈ 0c ↔ ∅ ∈ 0c)) |
18 | id 19 | . . . . . . . . . 10 ⊢ (g = (a ↑m ∅) → g = (a ↑m ∅)) | |
19 | oveq2 5531 | . . . . . . . . . 10 ⊢ (b = ∅ → (a ↑m b) = (a ↑m ∅)) | |
20 | 18, 19 | breqan12d 4654 | . . . . . . . . 9 ⊢ ((g = (a ↑m ∅) ∧ b = ∅) → (g ≈ (a ↑m b) ↔ (a ↑m ∅) ≈ (a ↑m ∅))) |
21 | 17, 20 | anbi12d 691 | . . . . . . . 8 ⊢ ((g = (a ↑m ∅) ∧ b = ∅) → ((℘1b ∈ 0c ∧ g ≈ (a ↑m b)) ↔ (∅ ∈ 0c ∧ (a ↑m ∅) ≈ (a ↑m ∅)))) |
22 | 10, 12, 21 | spc2ev 2947 | . . . . . . 7 ⊢ ((∅ ∈ 0c ∧ (a ↑m ∅) ≈ (a ↑m ∅)) → ∃g∃b(℘1b ∈ 0c ∧ g ≈ (a ↑m b))) |
23 | 9, 11, 22 | mp2an 653 | . . . . . 6 ⊢ ∃g∃b(℘1b ∈ 0c ∧ g ≈ (a ↑m b)) |
24 | 23 | biantru 491 | . . . . 5 ⊢ (℘1a ∈ M ↔ (℘1a ∈ M ∧ ∃g∃b(℘1b ∈ 0c ∧ g ≈ (a ↑m b)))) |
25 | 6, 8, 24 | 3bitr4ri 269 | . . . 4 ⊢ (℘1a ∈ M ↔ ∃g∃b(℘1a ∈ M ∧ ℘1b ∈ 0c ∧ g ≈ (a ↑m b))) |
26 | 25 | exbii 1582 | . . 3 ⊢ (∃a℘1a ∈ M ↔ ∃a∃g∃b(℘1a ∈ M ∧ ℘1b ∈ 0c ∧ g ≈ (a ↑m b))) |
27 | excom 1741 | . . 3 ⊢ (∃a∃g∃b(℘1a ∈ M ∧ ℘1b ∈ 0c ∧ g ≈ (a ↑m b)) ↔ ∃g∃a∃b(℘1a ∈ M ∧ ℘1b ∈ 0c ∧ g ≈ (a ↑m b))) | |
28 | 26, 27 | bitri 240 | . 2 ⊢ (∃a℘1a ∈ M ↔ ∃g∃a∃b(℘1a ∈ M ∧ ℘1b ∈ 0c ∧ g ≈ (a ↑m b))) |
29 | 4, 5, 28 | 3bitr4g 279 | 1 ⊢ (M ∈ NC → ((M ↑c 0c) ≠ ∅ ↔ ∃a℘1a ∈ M)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∧ w3a 934 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∅c0 3550 ℘1cpw1 4135 0cc0c 4374 class class class wbr 4639 (class class class)co 5525 ↑m cmap 5999 ≈ cen 6028 NC cncs 6088 ↑c cce 6096 |
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-ins2 5750 df-ins3 5752 df-image 5754 df-ins4 5756 df-si3 5758 df-funs 5760 df-fns 5762 df-pw1fn 5766 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: ce0nnuli 6178 ce0addcnnul 6179 ce0nnulb 6182 ceclb 6183 ce0nulnc 6184 ce0ncpw1 6185 te0c 6237 |
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