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Mirrors > Home > ILE Home > Th. List > 0nsr | GIF version |
Description: The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) |
Ref | Expression |
---|---|
0nsr | ⊢ ¬ ∅ ∈ R |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2100 | . 2 ⊢ ∅ = ∅ | |
2 | enrer 7431 | . . . . . 6 ⊢ ~R Er (P × P) | |
3 | erdm 6369 | . . . . . 6 ⊢ ( ~R Er (P × P) → dom ~R = (P × P)) | |
4 | 2, 3 | ax-mp 7 | . . . . 5 ⊢ dom ~R = (P × P) |
5 | elqsn0 6428 | . . . . 5 ⊢ ((dom ~R = (P × P) ∧ ∅ ∈ ((P × P) / ~R )) → ∅ ≠ ∅) | |
6 | 4, 5 | mpan 418 | . . . 4 ⊢ (∅ ∈ ((P × P) / ~R ) → ∅ ≠ ∅) |
7 | df-nr 7423 | . . . 4 ⊢ R = ((P × P) / ~R ) | |
8 | 6, 7 | eleq2s 2194 | . . 3 ⊢ (∅ ∈ R → ∅ ≠ ∅) |
9 | 8 | necon2bi 2322 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ R) |
10 | 1, 9 | ax-mp 7 | 1 ⊢ ¬ ∅ ∈ R |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1299 ∈ wcel 1448 ≠ wne 2267 ∅c0 3310 × cxp 4475 dom cdm 4477 Er wer 6356 / cqs 6358 Pcnp 7000 ~R cer 7005 Rcnr 7006 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-eprel 4149 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-irdg 6197 df-1o 6243 df-2o 6244 df-oadd 6247 df-omul 6248 df-er 6359 df-ec 6361 df-qs 6365 df-ni 7013 df-pli 7014 df-mi 7015 df-lti 7016 df-plpq 7053 df-mpq 7054 df-enq 7056 df-nqqs 7057 df-plqqs 7058 df-mqqs 7059 df-1nqqs 7060 df-rq 7061 df-ltnqqs 7062 df-enq0 7133 df-nq0 7134 df-0nq0 7135 df-plq0 7136 df-mq0 7137 df-inp 7175 df-iplp 7177 df-enr 7422 df-nr 7423 |
This theorem is referenced by: (None) |
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