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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 2727 | . 2 ⊢ ∅ = ∅ | |
2 | enrer 11072 | . . . . . 6 ⊢ ~R Er (P × P) | |
3 | erdm 8726 | . . . . . 6 ⊢ ( ~R Er (P × P) → dom ~R = (P × P)) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ dom ~R = (P × P) |
5 | elqsn0 8794 | . . . . 5 ⊢ ((dom ~R = (P × P) ∧ ∅ ∈ ((P × P) / ~R )) → ∅ ≠ ∅) | |
6 | 4, 5 | mpan 689 | . . . 4 ⊢ (∅ ∈ ((P × P) / ~R ) → ∅ ≠ ∅) |
7 | df-nr 11065 | . . . 4 ⊢ R = ((P × P) / ~R ) | |
8 | 6, 7 | eleq2s 2846 | . . 3 ⊢ (∅ ∈ R → ∅ ≠ ∅) |
9 | 8 | necon2bi 2966 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ R) |
10 | 1, 9 | ax-mp 5 | 1 ⊢ ¬ ∅ ∈ R |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∅c0 4318 × cxp 5670 dom cdm 5672 Er wer 8713 / cqs 8715 Pcnp 10868 ~R cer 10873 Rcnr 10874 |
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-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-oadd 8482 df-omul 8483 df-er 8716 df-ec 8718 df-qs 8722 df-ni 10881 df-pli 10882 df-mi 10883 df-lti 10884 df-plpq 10917 df-mpq 10918 df-ltpq 10919 df-enq 10920 df-nq 10921 df-erq 10922 df-plq 10923 df-mq 10924 df-1nq 10925 df-rq 10926 df-ltnq 10927 df-np 10990 df-plp 10992 df-ltp 10994 df-enr 11064 df-nr 11065 |
This theorem is referenced by: dmaddsr 11094 dmmulsr 11095 addasssr 11097 mulasssr 11099 distrsr 11100 ltasr 11109 |
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