Theorem renfdisj 8202

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 3540	. 2 ⊢ ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ∀𝑥 ∈ ℝ ¬ 𝑥 ∈ {+∞, -∞})
2		vex 2802	. . . . 5 ⊢ 𝑥 ∈ V
3	2	elpr 3687	. . . 4 ⊢ (𝑥 ∈ {+∞, -∞} ↔ (𝑥 = +∞ ∨ 𝑥 = -∞))
4		renepnf 8190	. . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ +∞)
5	4	necon2bi 2455	. . . . 5 ⊢ (𝑥 = +∞ → ¬ 𝑥 ∈ ℝ)
6		renemnf 8191	. . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ -∞)
7	6	necon2bi 2455	. . . . 5 ⊢ (𝑥 = -∞ → ¬ 𝑥 ∈ ℝ)
8	5, 7	jaoi 721	. . . 4 ⊢ ((𝑥 = +∞ ∨ 𝑥 = -∞) → ¬ 𝑥 ∈ ℝ)
9	3, 8	sylbi 121	. . 3 ⊢ (𝑥 ∈ {+∞, -∞} → ¬ 𝑥 ∈ ℝ)
10	9	con2i 630	. 2 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 ∈ {+∞, -∞})
11	1, 10	mprgbir 2588	1 ⊢ (ℝ ∩ {+∞, -∞}) = ∅

Syntax hints:  ¬ wn 3   ∨ wo 713   = wceq 1395   ∈ wcel 2200   ∩ cin 3196  ∅c0 3491  {cpr 3667  ℝcr 7994  +∞cpnf 8174  -∞cmnf 8175

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-uni 3888  df-pnf 8179  df-mnf 8180

This theorem is referenced by: (None)