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Mirrors > Home > ILE Home > Th. List > 0cnALT | GIF version |
Description: Alternate proof of 0cn 7997. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0cnALT | ⊢ 0 ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 7953 | . . 3 ⊢ i ∈ ℂ | |
2 | cnegex 8183 | . . 3 ⊢ (i ∈ ℂ → ∃𝑥 ∈ ℂ (i + 𝑥) = 0) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∃𝑥 ∈ ℂ (i + 𝑥) = 0 |
4 | addcl 7983 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i + 𝑥) ∈ ℂ) | |
5 | 1, 4 | mpan 424 | . . . 4 ⊢ (𝑥 ∈ ℂ → (i + 𝑥) ∈ ℂ) |
6 | eleq1 2252 | . . . 4 ⊢ ((i + 𝑥) = 0 → ((i + 𝑥) ∈ ℂ ↔ 0 ∈ ℂ)) | |
7 | 5, 6 | syl5ibcom 155 | . . 3 ⊢ (𝑥 ∈ ℂ → ((i + 𝑥) = 0 → 0 ∈ ℂ)) |
8 | 7 | rexlimiv 2601 | . 2 ⊢ (∃𝑥 ∈ ℂ (i + 𝑥) = 0 → 0 ∈ ℂ) |
9 | 3, 8 | ax-mp 5 | 1 ⊢ 0 ∈ ℂ |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 ∃wrex 2469 (class class class)co 5906 ℂcc 7856 0cc0 7858 ici 7860 + caddc 7861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-resscn 7950 ax-1cn 7951 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-addcom 7958 ax-addass 7960 ax-distr 7962 ax-i2m1 7963 ax-0id 7966 ax-rnegex 7967 ax-cnre 7969 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2758 df-un 3153 df-in 3155 df-ss 3162 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-br 4026 df-iota 5203 df-fv 5250 df-ov 5909 |
This theorem is referenced by: (None) |
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