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Mirrors > Home > MPE Home > Th. List > 0cnALT2 | Structured version Visualization version GIF version |
Description: Alternate proof of 0cnALT 10876 which is shorter, but depends on ax-8 2116, ax-13 2390, ax-sep 5205, ax-nul 5212, ax-pow 5268, ax-pr 5332, ax-un 7463, and every complex number axiom except ax-pre-mulgt0 10616 and ax-pre-sup 10617. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0cnALT2 | ⊢ 0 ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 10598 | . . 3 ⊢ i ∈ ℂ | |
2 | cnegex 10823 | . . 3 ⊢ (i ∈ ℂ → ∃𝑥 ∈ ℂ (i + 𝑥) = 0) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∃𝑥 ∈ ℂ (i + 𝑥) = 0 |
4 | addcl 10621 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i + 𝑥) ∈ ℂ) | |
5 | 1, 4 | mpan 688 | . . . 4 ⊢ (𝑥 ∈ ℂ → (i + 𝑥) ∈ ℂ) |
6 | eleq1 2902 | . . . 4 ⊢ ((i + 𝑥) = 0 → ((i + 𝑥) ∈ ℂ ↔ 0 ∈ ℂ)) | |
7 | 5, 6 | syl5ibcom 247 | . . 3 ⊢ (𝑥 ∈ ℂ → ((i + 𝑥) = 0 → 0 ∈ ℂ)) |
8 | 7 | rexlimiv 3282 | . 2 ⊢ (∃𝑥 ∈ ℂ (i + 𝑥) = 0 → 0 ∈ ℂ) |
9 | 3, 8 | ax-mp 5 | 1 ⊢ 0 ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∃wrex 3141 (class class class)co 7158 ℂcc 10537 0cc0 10539 ici 10541 + caddc 10542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 |
This theorem is referenced by: (None) |
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