0npr - Intuitionistic Logic Explorer

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Theorem 0npr 7816

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 3516	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2		1st0 6353	. . . . . . 7 ⊢ (1^st ‘∅) = ∅
3	2	eleq2i 2301	. . . . . 6 ⊢ (𝑥 ∈ (1^st ‘∅) ↔ 𝑥 ∈ ∅)
4	1, 3	mtbir 678	. . . . 5 ⊢ ¬ 𝑥 ∈ (1^st ‘∅)
5	4	nex 1549	. . . 4 ⊢ ¬ ∃𝑥 𝑥 ∈ (1^st ‘∅)
6		rexex 2590	. . . 4 ⊢ (∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅) → ∃𝑥 𝑥 ∈ (1^st ‘∅))
7	5, 6	mto 668	. . 3 ⊢ ¬ ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅)
8		prml 7810	. . 3 ⊢ (⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅))
9	7, 8	mto 668	. 2 ⊢ ¬ ⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P
10		prop 7808	. 2 ⊢ (∅ ∈ P → ⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P)
11	9, 10	mto 668	1 ⊢ ¬ ∅ ∈ P

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∃wex 1541 ∈ wcel 2205 ∃wrex 2523 ∅c0 3512 ⟨cop 3698 ‘cfv 5359 1^st c1st 6347 2^nd c2nd 6348 Qcnq 7613 Pcnp 7624

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2208 ax-ext 2216 ax-coll 4231 ax-sep 4234 ax-nul 4242 ax-pow 4293 ax-pr 4328 ax-un 4560 ax-iinf 4717

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-pw 3677 df-sn 3701 df-pr 3702 df-op 3704 df-uni 3921 df-int 3956 df-iun 3999 df-br 4116 df-opab 4178 df-mpt 4179 df-id 4420 df-iom 4720 df-xp 4762 df-rel 4763 df-cnv 4764 df-co 4765 df-dm 4766 df-rn 4767 df-res 4768 df-ima 4769 df-iota 5319 df-fun 5361 df-fn 5362 df-f 5363 df-f1 5364 df-fo 5365 df-f1o 5366 df-fv 5367 df-1st 6349 df-2nd 6350 df-qs 6788 df-ni 7637 df-nqqs 7681 df-inp 7799

This theorem is referenced by: (None)