0npr - Intuitionistic Logic Explorer

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Theorem 0npr 7596

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 3464	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2		1st0 6230	. . . . . . 7 ⊢ (1^st ‘∅) = ∅
3	2	eleq2i 2272	. . . . . 6 ⊢ (𝑥 ∈ (1^st ‘∅) ↔ 𝑥 ∈ ∅)
4	1, 3	mtbir 673	. . . . 5 ⊢ ¬ 𝑥 ∈ (1^st ‘∅)
5	4	nex 1523	. . . 4 ⊢ ¬ ∃𝑥 𝑥 ∈ (1^st ‘∅)
6		rexex 2552	. . . 4 ⊢ (∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅) → ∃𝑥 𝑥 ∈ (1^st ‘∅))
7	5, 6	mto 664	. . 3 ⊢ ¬ ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅)
8		prml 7590	. . 3 ⊢ (⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅))
9	7, 8	mto 664	. 2 ⊢ ¬ ⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P
10		prop 7588	. 2 ⊢ (∅ ∈ P → ⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P)
11	9, 10	mto 664	1 ⊢ ¬ ∅ ∈ P

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∃wex 1515 ∈ wcel 2176 ∃wrex 2485 ∅c0 3460 ⟨cop 3636 ‘cfv 5271 1^st c1st 6224 2^nd c2nd 6225 Qcnq 7393 Pcnp 7404

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-iinf 4636

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-1st 6226 df-2nd 6227 df-qs 6626 df-ni 7417 df-nqqs 7461 df-inp 7579

This theorem is referenced by: (None)