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Mirrors > Home > MPE Home > Th. List > 7re | Structured version Visualization version GIF version |
Description: The number 7 is real. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
7re | ⊢ 7 ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-7 11693 | . 2 ⊢ 7 = (6 + 1) | |
2 | 6re 11715 | . . 3 ⊢ 6 ∈ ℝ | |
3 | 1re 10629 | . . 3 ⊢ 1 ∈ ℝ | |
4 | 2, 3 | readdcli 10644 | . 2 ⊢ (6 + 1) ∈ ℝ |
5 | 1, 4 | eqeltri 2906 | 1 ⊢ 7 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 (class class class)co 7145 ℝcr 10524 1c1 10526 + caddc 10528 6c6 11684 7c7 11685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-i2m1 10593 ax-1ne0 10594 ax-rrecex 10597 ax-cnre 10598 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 |
This theorem is referenced by: 8re 11721 8pos 11737 5lt7 11812 4lt7 11813 3lt7 11814 2lt7 11815 1lt7 11816 7lt8 11817 6lt8 11818 7lt9 11825 6lt9 11826 7lt10 12219 6lt10 12220 bposlem8 25794 lgsdir2lem1 25828 hgt750lem2 31822 hgt750leme 31828 problem4 32808 mod42tp1mod8 43644 stgoldbwt 43818 sbgoldbwt 43819 nnsum3primesle9 43836 nnsum4primesoddALTV 43839 evengpoap3 43841 bgoldbtbndlem1 43847 bgoldbtbnd 43851 |
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