0npr - Intuitionistic Logic Explorer

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Theorem 0npr 7850

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 3525	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2		1st0 6378	. . . . . . 7 ⊢ (1^st ‘∅) = ∅
3	2	eleq2i 2305	. . . . . 6 ⊢ (𝑥 ∈ (1^st ‘∅) ↔ 𝑥 ∈ ∅)
4	1, 3	mtbir 682	. . . . 5 ⊢ ¬ 𝑥 ∈ (1^st ‘∅)
5	4	nex 1553	. . . 4 ⊢ ¬ ∃𝑥 𝑥 ∈ (1^st ‘∅)
6		rexex 2596	. . . 4 ⊢ (∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅) → ∃𝑥 𝑥 ∈ (1^st ‘∅))
7	5, 6	mto 672	. . 3 ⊢ ¬ ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅)
8		prml 7844	. . 3 ⊢ (⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘∅))
9	7, 8	mto 672	. 2 ⊢ ¬ ⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P
10		prop 7842	. 2 ⊢ (∅ ∈ P → ⟨(1^st ‘∅), (2^nd ‘∅)⟩ ∈ P)
11	9, 10	mto 672	1 ⊢ ¬ ∅ ∈ P

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 ∃wex 1545 ∈ wcel 2209 ∃wrex 2529 ∅c0 3520 ⟨cop 3712 ‘cfv 5377 1^st c1st 6372 2^nd c2nd 6373 Qcnq 7647 Pcnp 7658

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-iinf 4735

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-id 4438 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-1st 6374 df-2nd 6375 df-qs 6813 df-ni 7671 df-nqqs 7715 df-inp 7833

This theorem is used by: (None)

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