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Mirrors > Home > MPE Home > Th. List > 0cnALT2 | Structured version Visualization version GIF version |
Description: Alternate proof of 0cnALT 11063 which is shorter, but depends on ax-8 2112, ax-13 2371, ax-sep 5189, ax-nul 5196, ax-pow 5255, ax-pr 5319, ax-un 7520, and every complex number axiom except ax-pre-mulgt0 10803 and ax-pre-sup 10804. (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 10785 | . . 3 ⊢ i ∈ ℂ | |
2 | cnegex 11010 | . . 3 ⊢ (i ∈ ℂ → ∃𝑥 ∈ ℂ (i + 𝑥) = 0) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∃𝑥 ∈ ℂ (i + 𝑥) = 0 |
4 | addcl 10808 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝑥 ∈ ℂ) → (i + 𝑥) ∈ ℂ) | |
5 | 1, 4 | mpan 690 | . . . 4 ⊢ (𝑥 ∈ ℂ → (i + 𝑥) ∈ ℂ) |
6 | eleq1 2825 | . . . 4 ⊢ ((i + 𝑥) = 0 → ((i + 𝑥) ∈ ℂ ↔ 0 ∈ ℂ)) | |
7 | 5, 6 | syl5ibcom 248 | . . 3 ⊢ (𝑥 ∈ ℂ → ((i + 𝑥) = 0 → 0 ∈ ℂ)) |
8 | 7 | rexlimiv 3196 | . 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 1543 ∈ wcel 2110 ∃wrex 3059 (class class class)co 7210 ℂcc 10724 0cc0 10726 ici 10728 + caddc 10729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-po 5465 df-so 5466 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-ov 7213 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-pnf 10866 df-mnf 10867 df-ltxr 10869 |
This theorem is referenced by: (None) |
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