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Mirrors > Home > MPE Home > Th. List > addex | Structured version Visualization version GIF version |
Description: The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
addex | ⊢ + ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-addf 10015 | . 2 ⊢ + :(ℂ × ℂ)⟶ℂ | |
2 | cnex 10017 | . . 3 ⊢ ℂ ∈ V | |
3 | 2, 2 | xpex 6962 | . 2 ⊢ (ℂ × ℂ) ∈ V |
4 | fex2 7121 | . 2 ⊢ (( + :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → + ∈ V) | |
5 | 1, 3, 2, 4 | mp3an 1424 | 1 ⊢ + ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 Vcvv 3200 × cxp 5112 ⟶wf 5884 ℂcc 9934 + caddc 9939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-addf 10015 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 |
This theorem is referenced by: cnaddablx 18271 cnaddabl 18272 cnaddid 18273 cnaddinv 18274 zaddablx 18275 cnfldadd 19751 cnfldfun 19758 cnfldfunALT 19759 cnlmodlem2 22937 cnnvg 27533 cnnvs 27535 cncph 27674 cnaddcom 34259 |
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