Theorem renfdisj 8375

Description: The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Assertion

Ref	Expression
renfdisj	⊢ (ℝ ∩ {+∞, -∞}) = ∅

Proof of Theorem renfdisj

Step	Hyp	Ref	Expression
1		disj 3572	. 2 ⊢ ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ∀𝑥 ∈ ℝ ¬ 𝑥 ∈ {+∞, -∞})
2		vex 2824	. . . . 5 ⊢ 𝑥 ∈ V
3	2	elpr 3726	. . . 4 ⊢ (𝑥 ∈ {+∞, -∞} ↔ (𝑥 = +∞ ∨ 𝑥 = -∞))
4		renepnf 8363	. . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ +∞)
5	4	necon2bi 2475	. . . . 5 ⊢ (𝑥 = +∞ → ¬ 𝑥 ∈ ℝ)
6		renemnf 8364	. . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ -∞)
7	6	necon2bi 2475	. . . . 5 ⊢ (𝑥 = -∞ → ¬ 𝑥 ∈ ℝ)
8	5, 7	jaoi 728	. . . 4 ⊢ ((𝑥 = +∞ ∨ 𝑥 = -∞) → ¬ 𝑥 ∈ ℝ)
9	3, 8	sylbi 121	. . 3 ⊢ (𝑥 ∈ {+∞, -∞} → ¬ 𝑥 ∈ ℝ)
10	9	con2i 636	. 2 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 ∈ {+∞, -∞})
11	1, 10	mprgbir 2608	1 ⊢ (ℝ ∩ {+∞, -∞}) = ∅

Syntax hints:  ¬ wn 3   ∨ wo 720   = wceq 1402   ∈ wcel 2209   ∩ cin 3219  ∅c0 3520  {cpr 3706  ℝcr 8168  +∞cpnf 8347  -∞cmnf 8348

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 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-sep 4244  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-pnf 8352  df-mnf 8353

This theorem is referenced by: (None)