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Mirrors > Home > MPE Home > Th. List > nrex1 | Structured version Visualization version GIF version |
Description: The class of signed reals is a set. Note that a shorter proof is possible using qsex 8552 (and not requiring enrer 10829), but it would add a dependency on ax-rep 5208. (Contributed by Mario Carneiro, 17-Nov-2014.) Extract proof from that of axcnex 10913. (Revised by BJ, 4-Feb-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nrex1 | ⊢ R ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 10822 | . 2 ⊢ R = ((P × P) / ~R ) | |
2 | npex 10752 | . . . . 5 ⊢ P ∈ V | |
3 | 2, 2 | xpex 7593 | . . . 4 ⊢ (P × P) ∈ V |
4 | 3 | pwex 5301 | . . 3 ⊢ 𝒫 (P × P) ∈ V |
5 | enrer 10829 | . . . . . 6 ⊢ ~R Er (P × P) | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (⊤ → ~R Er (P × P)) |
7 | 6 | qsss 8554 | . . . 4 ⊢ (⊤ → ((P × P) / ~R ) ⊆ 𝒫 (P × P)) |
8 | 7 | mptru 1546 | . . 3 ⊢ ((P × P) / ~R ) ⊆ 𝒫 (P × P) |
9 | 4, 8 | ssexi 5244 | . 2 ⊢ ((P × P) / ~R ) ∈ V |
10 | 1, 9 | eqeltri 2835 | 1 ⊢ R ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1540 ∈ wcel 2106 Vcvv 3429 ⊆ wss 3886 𝒫 cpw 4533 × cxp 5582 Er wer 8482 / cqs 8484 Pcnp 10625 ~R cer 10630 Rcnr 10631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-inf2 9386 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-oadd 8288 df-omul 8289 df-er 8485 df-ec 8487 df-qs 8491 df-ni 10638 df-pli 10639 df-mi 10640 df-lti 10641 df-plpq 10674 df-mpq 10675 df-ltpq 10676 df-enq 10677 df-nq 10678 df-erq 10679 df-plq 10680 df-mq 10681 df-1nq 10682 df-rq 10683 df-ltnq 10684 df-np 10747 df-plp 10749 df-ltp 10751 df-enr 10821 df-nr 10822 |
This theorem is referenced by: axcnex 10913 bj-inftyexpitaudisj 35384 |
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