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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 2738 | . 2 ⊢ ∅ = ∅ | |
2 | enrer 10829 | . . . . . 6 ⊢ ~R Er (P × P) | |
3 | erdm 8495 | . . . . . 6 ⊢ ( ~R Er (P × P) → dom ~R = (P × P)) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ dom ~R = (P × P) |
5 | elqsn0 8562 | . . . . 5 ⊢ ((dom ~R = (P × P) ∧ ∅ ∈ ((P × P) / ~R )) → ∅ ≠ ∅) | |
6 | 4, 5 | mpan 687 | . . . 4 ⊢ (∅ ∈ ((P × P) / ~R ) → ∅ ≠ ∅) |
7 | df-nr 10822 | . . . 4 ⊢ R = ((P × P) / ~R ) | |
8 | 6, 7 | eleq2s 2857 | . . 3 ⊢ (∅ ∈ R → ∅ ≠ ∅) |
9 | 8 | necon2bi 2974 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ R) |
10 | 1, 9 | ax-mp 5 | 1 ⊢ ¬ ∅ ∈ R |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∅c0 4256 × cxp 5582 dom cdm 5584 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: dmaddsr 10851 dmmulsr 10852 addasssr 10854 mulasssr 10856 distrsr 10857 ltasr 10866 |
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