0npr - Intuitionistic Logic Explorer

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Theorem 0npr 7567

Description: The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)

**Assertion**
Ref	Expression
0npr	⊢ ¬ ∅ ∈ P

**Proof of Theorem 0npr**
Step	Hyp	Ref	Expression
1		noel 3455	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2		1st0 6211	. . . . . . 7 ⊢ (1^st ‘∅) = ∅
3	2	eleq2i 2263	. . . . . 6 ⊢ (𝑥 ∈ (1^st ‘∅) ↔ 𝑥 ∈ ∅)
4	1, 3	mtbir 672	. . . . 5 ⊢ ¬ 𝑥 ∈ (1^st ‘∅)
5	4	nex 1514	. . . 4 ⊢ ¬ ∃𝑥 𝑥 ∈ (1^st ‘∅)
6		rexex 2543	. . . 4 ⊢ (∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅) → ∃𝑥 𝑥 ∈ (1^st ‘∅))
7	5, 6	mto 663	. . 3 ⊢ ¬ ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅)
8		prml 7561	. . 3 ⊢ (⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅))
9	7, 8	mto 663	. 2 ⊢ ¬ ⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P
10		prop 7559	. 2 ⊢ (∅ ∈ P → ⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P)
11	9, 10	mto 663	1 ⊢ ¬ ∅ ∈ P

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∃wex 1506 ∈ wcel 2167 ∃wrex 2476 ∅c0 3451 ⟨cop 3626 ‘cfv 5259 1^st c1st 6205 2^nd c2nd 6206 Qcnq 7364 Pcnp 7375

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-iinf 4625

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-id 4329 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-1st 6207 df-2nd 6208 df-qs 6607 df-ni 7388 df-nqqs 7432 df-inp 7550

This theorem is referenced by: (None)