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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 11237 | . 2 ⊢ + :(ℂ × ℂ)⟶ℂ | |
2 | cnex 11239 | . . 3 ⊢ ℂ ∈ V | |
3 | 2, 2 | xpex 7761 | . 2 ⊢ (ℂ × ℂ) ∈ V |
4 | fex2 7947 | . 2 ⊢ (( + :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → + ∈ V) | |
5 | 1, 3, 2, 4 | mp3an 1458 | 1 ⊢ + ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 Vcvv 3462 × cxp 5680 ⟶wf 6550 ℂcc 11156 + caddc 11161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-addf 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-xp 5688 df-rel 5689 df-cnv 5690 df-dm 5692 df-rn 5693 df-fun 6556 df-fn 6557 df-f 6558 |
This theorem is referenced by: cnaddablx 19866 cnaddabl 19867 cnaddid 19868 cnaddinv 19869 zaddablx 19870 cnfldaddOLD 21363 cnfldfunOLD 21370 cnfldfunALTOLD 21371 cnfldfunALTOLDOLD 21372 cnlmodlem2 25155 cnnvg 30611 cnnvs 30613 cncph 30752 cnaddcom 38670 nn0mnd 47556 |
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