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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 11129 | . 2 ⊢ + :(ℂ × ℂ)⟶ℂ | |
2 | cnex 11131 | . . 3 ⊢ ℂ ∈ V | |
3 | 2, 2 | xpex 7686 | . 2 ⊢ (ℂ × ℂ) ∈ V |
4 | fex2 7869 | . 2 ⊢ (( + :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → + ∈ V) | |
5 | 1, 3, 2, 4 | mp3an 1461 | 1 ⊢ + ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3445 × cxp 5631 ⟶wf 6492 ℂcc 11048 + caddc 11053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-addf 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-xp 5639 df-rel 5640 df-cnv 5641 df-dm 5643 df-rn 5644 df-fun 6498 df-fn 6499 df-f 6500 |
This theorem is referenced by: cnaddablx 19644 cnaddabl 19645 cnaddid 19646 cnaddinv 19647 zaddablx 19648 cnfldadd 20799 cnfldfun 20806 cnfldfunALT 20807 cnfldfunALTOLD 20808 cnlmodlem2 24498 cnnvg 29618 cnnvs 29620 cncph 29759 cnaddcom 37425 nn0mnd 46085 |
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