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Mirrors > Home > ILE Home > Th. List > 0cnALT | GIF version |
Description: Alternate proof of 0cn 7758. (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 7715 | . . 3 ⊢ i ∈ ℂ | |
2 | cnegex 7940 | . . 3 ⊢ (i ∈ ℂ → ∃𝑥 ∈ ℂ (i + 𝑥) = 0) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∃𝑥 ∈ ℂ (i + 𝑥) = 0 |
4 | addcl 7745 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i + 𝑥) ∈ ℂ) | |
5 | 1, 4 | mpan 420 | . . . 4 ⊢ (𝑥 ∈ ℂ → (i + 𝑥) ∈ ℂ) |
6 | eleq1 2202 | . . . 4 ⊢ ((i + 𝑥) = 0 → ((i + 𝑥) ∈ ℂ ↔ 0 ∈ ℂ)) | |
7 | 5, 6 | syl5ibcom 154 | . . 3 ⊢ (𝑥 ∈ ℂ → ((i + 𝑥) = 0 → 0 ∈ ℂ)) |
8 | 7 | rexlimiv 2543 | . 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 1331 ∈ wcel 1480 ∃wrex 2417 (class class class)co 5774 ℂcc 7618 0cc0 7620 ici 7622 + caddc 7623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-resscn 7712 ax-1cn 7713 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 |
This theorem is referenced by: (None) |
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