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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 8710 (and not requiring enrer 10978), but it would add a dependency on ax-rep 5200. (Contributed by Mario Carneiro, 17-Nov-2014.) Extract proof from that of axcnex 11062. (Revised by BJ, 4-Feb-2023.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nrex1 | ⊢ R ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-nr 10971 | . 2 ⊢ R = ((P × P) / ~R ) | |
| 2 | npex 10901 | . . . . 5 ⊢ P ∈ V | |
| 3 | 2, 2 | xpex 7697 | . . . 4 ⊢ (P × P) ∈ V |
| 4 | 3 | pwex 5310 | . . 3 ⊢ 𝒫 (P × P) ∈ V |
| 5 | enrer 10978 | . . . . . 6 ⊢ ~R Er (P × P) | |
| 6 | 5 | a1i 11 | . . . . 5 ⊢ (⊤ → ~R Er (P × P)) |
| 7 | 6 | qsss 8713 | . . . 4 ⊢ (⊤ → ((P × P) / ~R ) ⊆ 𝒫 (P × P)) |
| 8 | 7 | mptru 1554 | . . 3 ⊢ ((P × P) / ~R ) ⊆ 𝒫 (P × P) |
| 9 | 4, 8 | ssexi 5251 | . 2 ⊢ ((P × P) / ~R ) ∈ V |
| 10 | 1, 9 | eqeltri 2835 | 1 ⊢ R ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ⊤wtru 1548 ∈ wcel 2119 Vcvv 3431 ⊆ wss 3883 𝒫 cpw 4530 × cxp 5617 Er wer 8631 / cqs 8633 Pcnp 10774 ~R cer 10779 Rcnr 10780 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-inf2 9554 |
| This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7808 df-1st 7932 df-2nd 7933 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-oadd 8400 df-omul 8401 df-er 8634 df-ec 8636 df-qs 8640 df-ni 10787 df-pli 10788 df-mi 10789 df-lti 10790 df-plpq 10823 df-mpq 10824 df-ltpq 10825 df-enq 10826 df-nq 10827 df-erq 10828 df-plq 10829 df-mq 10830 df-1nq 10831 df-rq 10832 df-ltnq 10833 df-np 10896 df-plp 10898 df-ltp 10900 df-enr 10970 df-nr 10971 |
| This theorem is referenced by: axcnex 11062 bj-inftyexpitaudisj 37574 |
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