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Mirrors > Home > ILE Home > Th. List > 0npr | GIF version |
Description: The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) |
Ref | Expression |
---|---|
0npr | ⊢ ¬ ∅ ∈ P |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3255 | . . . . . 6 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1st0 5791 | . . . . . . 7 ⊢ (1st ‘∅) = ∅ | |
3 | 2 | eleq2i 2145 | . . . . . 6 ⊢ (𝑥 ∈ (1st ‘∅) ↔ 𝑥 ∈ ∅) |
4 | 1, 3 | mtbir 628 | . . . . 5 ⊢ ¬ 𝑥 ∈ (1st ‘∅) |
5 | 4 | nex 1429 | . . . 4 ⊢ ¬ ∃𝑥 𝑥 ∈ (1st ‘∅) |
6 | rexex 2410 | . . . 4 ⊢ (∃𝑥 ∈ Q 𝑥 ∈ (1st ‘∅) → ∃𝑥 𝑥 ∈ (1st ‘∅)) | |
7 | 5, 6 | mto 620 | . . 3 ⊢ ¬ ∃𝑥 ∈ Q 𝑥 ∈ (1st ‘∅) |
8 | prml 6667 | . . 3 ⊢ (〈(1st ‘∅), (2nd ‘∅)〉 ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1st ‘∅)) | |
9 | 7, 8 | mto 620 | . 2 ⊢ ¬ 〈(1st ‘∅), (2nd ‘∅)〉 ∈ P |
10 | prop 6665 | . 2 ⊢ (∅ ∈ P → 〈(1st ‘∅), (2nd ‘∅)〉 ∈ P) | |
11 | 9, 10 | mto 620 | 1 ⊢ ¬ ∅ ∈ P |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∃wex 1421 ∈ wcel 1433 ∃wrex 2349 ∅c0 3251 〈cop 3401 ‘cfv 4922 1st c1st 5785 2nd c2nd 5786 Qcnq 6470 Pcnp 6481 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-1st 5787 df-2nd 5788 df-qs 6135 df-ni 6494 df-nqqs 6538 df-inp 6656 |
This theorem is referenced by: (None) |
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