0npr - Intuitionistic Logic Explorer

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Theorem 0npr 7484

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 3428	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2		1st0 6147	. . . . . . 7 ⊢ (1^st ‘∅) = ∅
3	2	eleq2i 2244	. . . . . 6 ⊢ (𝑥 ∈ (1^st ‘∅) ↔ 𝑥 ∈ ∅)
4	1, 3	mtbir 671	. . . . 5 ⊢ ¬ 𝑥 ∈ (1^st ‘∅)
5	4	nex 1500	. . . 4 ⊢ ¬ ∃𝑥 𝑥 ∈ (1^st ‘∅)
6		rexex 2523	. . . 4 ⊢ (∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅) → ∃𝑥 𝑥 ∈ (1^st ‘∅))
7	5, 6	mto 662	. . 3 ⊢ ¬ ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅)
8		prml 7478	. . 3 ⊢ (⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅))
9	7, 8	mto 662	. 2 ⊢ ¬ ⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P
10		prop 7476	. 2 ⊢ (∅ ∈ P → ⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P)
11	9, 10	mto 662	1 ⊢ ¬ ∅ ∈ P

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∃wex 1492 ∈ wcel 2148 ∃wrex 2456 ∅c0 3424 ⟨cop 3597 ‘cfv 5218 1^st c1st 6141 2^nd c2nd 6142 Qcnq 7281 Pcnp 7292

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-iinf 4589

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-1st 6143 df-2nd 6144 df-qs 6543 df-ni 7305 df-nqqs 7349 df-inp 7467

This theorem is referenced by: (None)