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Mirrors > Home > ILE Home > Th. List > renepnf | GIF version |
Description: No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
renepnf | ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfnre 7948 | . . . 4 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 2437 | . . 3 ⊢ ¬ +∞ ∈ ℝ |
3 | eleq1 2233 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 ∈ ℝ ↔ +∞ ∈ ℝ)) | |
4 | 2, 3 | mtbiri 670 | . 2 ⊢ (𝐴 = +∞ → ¬ 𝐴 ∈ ℝ) |
5 | 4 | necon2ai 2394 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ℝcr 7760 +∞cpnf 7938 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-un 4416 ax-cnex 7852 ax-resscn 7853 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-rex 2454 df-rab 2457 df-v 2732 df-in 3127 df-ss 3134 df-pw 3566 df-uni 3795 df-pnf 7943 |
This theorem is referenced by: renepnfd 7957 renfdisj 7966 ltxrlt 7972 xrnepnf 9722 xrlttri3 9741 nltpnft 9758 xrrebnd 9763 rexneg 9774 xrpnfdc 9786 rexadd 9796 xaddnepnf 9802 xaddcom 9805 xaddid1 9806 xnn0xadd0 9811 xnegdi 9812 xpncan 9815 xleadd1a 9817 xltadd1 9820 xsubge0 9825 xposdif 9826 xleaddadd 9831 xrmaxrecl 11205 |
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