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Mirrors > Home > NFE Home > Th. List > addcid1 | GIF version |
Description: Cardinal zero is a fixed point for cardinal addition. Theorem X.1.8 of [Rosser] p. 276. (Contributed by SF, 16-Jan-2015.) |
Ref | Expression |
---|---|
addcid1 | ⊢ (A +c 0c) = A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-0c 4377 | . . 3 ⊢ 0c = {∅} | |
2 | 1 | addceq2i 4387 | . 2 ⊢ (A +c 0c) = (A +c {∅}) |
3 | 0ex 4110 | . . . . . . 7 ⊢ ∅ ∈ V | |
4 | ineq2 3451 | . . . . . . . . . 10 ⊢ (z = ∅ → (y ∩ z) = (y ∩ ∅)) | |
5 | 4 | eqeq1d 2361 | . . . . . . . . 9 ⊢ (z = ∅ → ((y ∩ z) = ∅ ↔ (y ∩ ∅) = ∅)) |
6 | uneq2 3412 | . . . . . . . . . 10 ⊢ (z = ∅ → (y ∪ z) = (y ∪ ∅)) | |
7 | 6 | eqeq2d 2364 | . . . . . . . . 9 ⊢ (z = ∅ → (x = (y ∪ z) ↔ x = (y ∪ ∅))) |
8 | 5, 7 | anbi12d 691 | . . . . . . . 8 ⊢ (z = ∅ → (((y ∩ z) = ∅ ∧ x = (y ∪ z)) ↔ ((y ∩ ∅) = ∅ ∧ x = (y ∪ ∅)))) |
9 | in0 3576 | . . . . . . . . 9 ⊢ (y ∩ ∅) = ∅ | |
10 | 9 | biantrur 492 | . . . . . . . 8 ⊢ (x = (y ∪ ∅) ↔ ((y ∩ ∅) = ∅ ∧ x = (y ∪ ∅))) |
11 | 8, 10 | syl6bbr 254 | . . . . . . 7 ⊢ (z = ∅ → (((y ∩ z) = ∅ ∧ x = (y ∪ z)) ↔ x = (y ∪ ∅))) |
12 | 3, 11 | rexsn 3768 | . . . . . 6 ⊢ (∃z ∈ {∅} ((y ∩ z) = ∅ ∧ x = (y ∪ z)) ↔ x = (y ∪ ∅)) |
13 | un0 3575 | . . . . . . 7 ⊢ (y ∪ ∅) = y | |
14 | 13 | eqeq2i 2363 | . . . . . 6 ⊢ (x = (y ∪ ∅) ↔ x = y) |
15 | equcom 1680 | . . . . . 6 ⊢ (x = y ↔ y = x) | |
16 | 12, 14, 15 | 3bitri 262 | . . . . 5 ⊢ (∃z ∈ {∅} ((y ∩ z) = ∅ ∧ x = (y ∪ z)) ↔ y = x) |
17 | 16 | rexbii 2639 | . . . 4 ⊢ (∃y ∈ A ∃z ∈ {∅} ((y ∩ z) = ∅ ∧ x = (y ∪ z)) ↔ ∃y ∈ A y = x) |
18 | eladdc 4398 | . . . 4 ⊢ (x ∈ (A +c {∅}) ↔ ∃y ∈ A ∃z ∈ {∅} ((y ∩ z) = ∅ ∧ x = (y ∪ z))) | |
19 | risset 2661 | . . . 4 ⊢ (x ∈ A ↔ ∃y ∈ A y = x) | |
20 | 17, 18, 19 | 3bitr4i 268 | . . 3 ⊢ (x ∈ (A +c {∅}) ↔ x ∈ A) |
21 | 20 | eqriv 2350 | . 2 ⊢ (A +c {∅}) = A |
22 | 2, 21 | eqtri 2373 | 1 ⊢ (A +c 0c) = A |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 ∪ cun 3207 ∩ cin 3208 ∅c0 3550 {csn 3737 0cc0c 4374 +c cplc 4375 |
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-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-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 |
This theorem is referenced by: addcid2 4407 1cnnc 4408 nncaddccl 4419 ltfinirr 4457 ltfinp1 4462 lefinlteq 4463 lefinrflx 4467 vfin1cltv 4547 nclenn 6249 ncslesuc 6267 nncdiv3 6277 nnc3n3p1 6278 nchoicelem17 6305 |
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