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Mirrors > Home > MPE Home > Th. List > renfdisj | Structured version Visualization version GIF version |
Description: The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
renfdisj | ⊢ (ℝ ∩ {+∞, -∞}) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disj 4408 | . 2 ⊢ ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ∀𝑥 ∈ ℝ ¬ 𝑥 ∈ {+∞, -∞}) | |
2 | renepnf 11208 | . . 3 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ +∞) | |
3 | renemnf 11209 | . . 3 ⊢ (𝑥 ∈ ℝ → 𝑥 ≠ -∞) | |
4 | 2, 3 | nelprd 4618 | . 2 ⊢ (𝑥 ∈ ℝ → ¬ 𝑥 ∈ {+∞, -∞}) |
5 | 1, 4 | mprgbir 3068 | 1 ⊢ (ℝ ∩ {+∞, -∞}) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 ∩ cin 3910 ∅c0 4283 {cpr 4589 ℝcr 11055 +∞cpnf 11191 -∞cmnf 11192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11113 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 |
This theorem is referenced by: ssxr 11229 |
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