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Theorem 0npr 7259
Description: The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
Assertion
Ref Expression
0npr ¬ ∅ ∈ P

Proof of Theorem 0npr
StepHypRef Expression
1 noel 3337 . . . . . 6 ¬ 𝑥 ∈ ∅
2 1st0 6010 . . . . . . 7 (1st ‘∅) = ∅
32eleq2i 2184 . . . . . 6 (𝑥 ∈ (1st ‘∅) ↔ 𝑥 ∈ ∅)
41, 3mtbir 645 . . . . 5 ¬ 𝑥 ∈ (1st ‘∅)
54nex 1461 . . . 4 ¬ ∃𝑥 𝑥 ∈ (1st ‘∅)
6 rexex 2456 . . . 4 (∃𝑥Q 𝑥 ∈ (1st ‘∅) → ∃𝑥 𝑥 ∈ (1st ‘∅))
75, 6mto 636 . . 3 ¬ ∃𝑥Q 𝑥 ∈ (1st ‘∅)
8 prml 7253 . . 3 (⟨(1st ‘∅), (2nd ‘∅)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (1st ‘∅))
97, 8mto 636 . 2 ¬ ⟨(1st ‘∅), (2nd ‘∅)⟩ ∈ P
10 prop 7251 . 2 (∅ ∈ P → ⟨(1st ‘∅), (2nd ‘∅)⟩ ∈ P)
119, 10mto 636 1 ¬ ∅ ∈ P
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wex 1453  wcel 1465  wrex 2394  c0 3333  cop 3500  cfv 5093  1st c1st 6004  2nd c2nd 6005  Qcnq 7056  Pcnp 7067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-qs 6403  df-ni 7080  df-nqqs 7124  df-inp 7242
This theorem is referenced by: (None)
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