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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 2826 | . 2 ⊢ ∅ = ∅ | |
2 | enrer 10201 | . . . . . 6 ⊢ ~R Er (P × P) | |
3 | erdm 8020 | . . . . . 6 ⊢ ( ~R Er (P × P) → dom ~R = (P × P)) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ dom ~R = (P × P) |
5 | elqsn0 8082 | . . . . 5 ⊢ ((dom ~R = (P × P) ∧ ∅ ∈ ((P × P) / ~R )) → ∅ ≠ ∅) | |
6 | 4, 5 | mpan 683 | . . . 4 ⊢ (∅ ∈ ((P × P) / ~R ) → ∅ ≠ ∅) |
7 | df-nr 10194 | . . . 4 ⊢ R = ((P × P) / ~R ) | |
8 | 6, 7 | eleq2s 2925 | . . 3 ⊢ (∅ ∈ R → ∅ ≠ ∅) |
9 | 8 | necon2bi 3030 | . 2 ⊢ (∅ = ∅ → ¬ ∅ ∈ R) |
10 | 1, 9 | ax-mp 5 | 1 ⊢ ¬ ∅ ∈ R |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 ∅c0 4145 × cxp 5341 dom cdm 5343 Er wer 8007 / cqs 8009 Pcnp 9997 ~R cer 10002 Rcnr 10003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-inf2 8816 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-omul 7832 df-er 8010 df-ec 8012 df-qs 8016 df-ni 10010 df-pli 10011 df-mi 10012 df-lti 10013 df-plpq 10046 df-mpq 10047 df-ltpq 10048 df-enq 10049 df-nq 10050 df-erq 10051 df-plq 10052 df-mq 10053 df-1nq 10054 df-rq 10055 df-ltnq 10056 df-np 10119 df-plp 10121 df-ltp 10123 df-enr 10193 df-nr 10194 |
This theorem is referenced by: dmaddsr 10223 dmmulsr 10224 addasssr 10226 mulasssr 10228 distrsr 10229 ltasr 10238 |
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