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Mirrors > Home > MPE Home > Th. List > 0nsr | Structured version Visualization version GIF version |
Description: The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0nsr | ⊢ ¬ ∅ ∈ R |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . 2 ⊢ ∅ = ∅ | |
2 | enrer 10484 | . . . . . 6 ⊢ ~R Er (P × P) | |
3 | erdm 8298 | . . . . . 6 ⊢ ( ~R Er (P × P) → dom ~R = (P × P)) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ dom ~R = (P × P) |
5 | elqsn0 8365 | . . . . 5 ⊢ ((dom ~R = (P × P) ∧ ∅ ∈ ((P × P) / ~R )) → ∅ ≠ ∅) | |
6 | 4, 5 | mpan 688 | . . . 4 ⊢ (∅ ∈ ((P × P) / ~R ) → ∅ ≠ ∅) |
7 | df-nr 10477 | . . . 4 ⊢ R = ((P × P) / ~R ) | |
8 | 6, 7 | eleq2s 2931 | . . 3 ⊢ (∅ ∈ R → ∅ ≠ ∅) |
9 | 8 | necon2bi 3046 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ R) |
10 | 1, 9 | ax-mp 5 | 1 ⊢ ¬ ∅ ∈ R |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∅c0 4290 × cxp 5552 dom cdm 5554 Er wer 8285 / cqs 8287 Pcnp 10280 ~R cer 10285 Rcnr 10286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-omul 8106 df-er 8288 df-ec 8290 df-qs 8294 df-ni 10293 df-pli 10294 df-mi 10295 df-lti 10296 df-plpq 10329 df-mpq 10330 df-ltpq 10331 df-enq 10332 df-nq 10333 df-erq 10334 df-plq 10335 df-mq 10336 df-1nq 10337 df-rq 10338 df-ltnq 10339 df-np 10402 df-plp 10404 df-ltp 10406 df-enr 10476 df-nr 10477 |
This theorem is referenced by: dmaddsr 10506 dmmulsr 10507 addasssr 10509 mulasssr 10511 distrsr 10512 ltasr 10521 |
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