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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 11381 | . 2 ⊢ 7 = (6 + 1) | |
2 | 6re 11406 | . . 3 ⊢ 6 ∈ ℝ | |
3 | 1re 10328 | . . 3 ⊢ 1 ∈ ℝ | |
4 | 2, 3 | readdcli 10344 | . 2 ⊢ (6 + 1) ∈ ℝ |
5 | 1, 4 | eqeltri 2874 | 1 ⊢ 7 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2157 (class class class)co 6878 ℝcr 10223 1c1 10225 + caddc 10227 6c6 11372 7c7 11373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-i2m1 10292 ax-1ne0 10293 ax-rrecex 10296 ax-cnre 10297 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-iota 6064 df-fv 6109 df-ov 6881 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 |
This theorem is referenced by: 7cnOLD 11412 8re 11414 8pos 11432 5lt7 11507 4lt7 11508 3lt7 11509 2lt7 11510 1lt7 11511 7lt8 11512 6lt8 11513 7lt9 11520 6lt9 11521 7lt10 11918 6lt10 11919 bposlem8 25368 lgsdir2lem1 25402 hgt750lem2 31250 hgt750leme 31256 problem4 32077 mod42tp1mod8 42301 stgoldbwt 42446 sbgoldbwt 42447 nnsum3primesle9 42464 nnsum4primesoddALTV 42467 evengpoap3 42469 bgoldbtbndlem1 42475 bgoldbtbnd 42479 |
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