Theorem 0npr 7702

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 3498	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
2		1st0 6306	. . . . . . 7 ⊢ (1st ‘∅) = ∅
3	2	eleq2i 2298	. . . . . 6 ⊢ (𝑥 ∈ (1st ‘∅) ↔ 𝑥 ∈ ∅)
4	1, 3	mtbir 677	. . . . 5 ⊢ ¬ 𝑥 ∈ (1st ‘∅)
5	4	nex 1548	. . . 4 ⊢ ¬ ∃𝑥 𝑥 ∈ (1st ‘∅)
6		rexex 2578	. . . 4 ⊢ (∃𝑥 ∈ Q 𝑥 ∈ (1st ‘∅) → ∃𝑥 𝑥 ∈ (1st ‘∅))
7	5, 6	mto 668	. . 3 ⊢ ¬ ∃𝑥 ∈ Q 𝑥 ∈ (1st ‘∅)
8		prml 7696	. . 3 ⊢ (⟨(1st ‘∅), (2nd ‘∅)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1st ‘∅))
9	7, 8	mto 668	. 2 ⊢ ¬ ⟨(1st ‘∅), (2nd ‘∅)⟩ ∈ P
10		prop 7694	. 2 ⊢ (∅ ∈ P → ⟨(1st ‘∅), (2nd ‘∅)⟩ ∈ P)
11	9, 10	mto 668	1 ⊢ ¬ ∅ ∈ P

Syntax hints:  ¬ wn 3  ∃wex 1540   ∈ wcel 2202  ∃wrex 2511  ∅c0 3494  ⟨cop 3672  ‘cfv 5326  1^st c1st 6300  2^nd c2nd 6301  Qcnq 7499  Pcnp 7510

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-qs 6707  df-ni 7523  df-nqqs 7567  df-inp 7685

This theorem is referenced by: (None)