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Mirrors > Home > NFE Home > Th. List > 0ceven | GIF version |
Description: Cardinal zero is even. (Contributed by SF, 20-Jan-2015.) |
Ref | Expression |
---|---|
0ceven | ⊢ 0c ∈ Evenfin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 4402 | . . 3 ⊢ 0c ∈ Nn | |
2 | addcid2 4407 | . . . 4 ⊢ (0c +c 0c) = 0c | |
3 | 2 | eqcomi 2357 | . . 3 ⊢ 0c = (0c +c 0c) |
4 | addceq12 4385 | . . . . . 6 ⊢ ((n = 0c ∧ n = 0c) → (n +c n) = (0c +c 0c)) | |
5 | 4 | anidms 626 | . . . . 5 ⊢ (n = 0c → (n +c n) = (0c +c 0c)) |
6 | 5 | eqeq2d 2364 | . . . 4 ⊢ (n = 0c → (0c = (n +c n) ↔ 0c = (0c +c 0c))) |
7 | 6 | rspcev 2955 | . . 3 ⊢ ((0c ∈ Nn ∧ 0c = (0c +c 0c)) → ∃n ∈ Nn 0c = (n +c n)) |
8 | 1, 3, 7 | mp2an 653 | . 2 ⊢ ∃n ∈ Nn 0c = (n +c n) |
9 | 0ex 4110 | . . . . 5 ⊢ ∅ ∈ V | |
10 | 9 | snid 3760 | . . . 4 ⊢ ∅ ∈ {∅} |
11 | df-0c 4377 | . . . 4 ⊢ 0c = {∅} | |
12 | 10, 11 | eleqtrri 2426 | . . 3 ⊢ ∅ ∈ 0c |
13 | ne0i 3556 | . . 3 ⊢ (∅ ∈ 0c → 0c ≠ ∅) | |
14 | 12, 13 | ax-mp 5 | . 2 ⊢ 0c ≠ ∅ |
15 | 0cex 4392 | . . 3 ⊢ 0c ∈ V | |
16 | eqeq1 2359 | . . . . 5 ⊢ (x = 0c → (x = (n +c n) ↔ 0c = (n +c n))) | |
17 | 16 | rexbidv 2635 | . . . 4 ⊢ (x = 0c → (∃n ∈ Nn x = (n +c n) ↔ ∃n ∈ Nn 0c = (n +c n))) |
18 | neeq1 2524 | . . . 4 ⊢ (x = 0c → (x ≠ ∅ ↔ 0c ≠ ∅)) | |
19 | 17, 18 | anbi12d 691 | . . 3 ⊢ (x = 0c → ((∃n ∈ Nn x = (n +c n) ∧ x ≠ ∅) ↔ (∃n ∈ Nn 0c = (n +c n) ∧ 0c ≠ ∅))) |
20 | df-evenfin 4444 | . . 3 ⊢ Evenfin = {x ∣ (∃n ∈ Nn x = (n +c n) ∧ x ≠ ∅)} | |
21 | 15, 19, 20 | elab2 2988 | . 2 ⊢ (0c ∈ Evenfin ↔ (∃n ∈ Nn 0c = (n +c n) ∧ 0c ≠ ∅)) |
22 | 8, 14, 21 | mpbir2an 886 | 1 ⊢ 0c ∈ Evenfin |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∃wrex 2615 ∅c0 3550 {csn 3737 Nn cnnc 4373 0cc0c 4374 +c cplc 4375 Evenfin cevenfin 4436 |
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-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 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-nul 3551 df-pw 3724 df-sn 3741 df-pr 3742 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-sik 4192 df-ssetk 4193 df-0c 4377 df-addc 4378 df-nnc 4379 df-evenfin 4444 |
This theorem is referenced by: evenoddnnnul 4514 |
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