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Mirrors > Home > MPE Home > Th. List > 1pr | Structured version Visualization version GIF version |
Description: The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1pr | ⊢ 1P ∈ P |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1p 10406 | . 2 ⊢ 1P = {𝑥 ∣ 𝑥 <Q 1Q} | |
2 | 1nq 10352 | . . 3 ⊢ 1Q ∈ Q | |
3 | nqpr 10438 | . . 3 ⊢ (1Q ∈ Q → {𝑥 ∣ 𝑥 <Q 1Q} ∈ P) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ {𝑥 ∣ 𝑥 <Q 1Q} ∈ P |
5 | 1, 4 | eqeltri 2911 | 1 ⊢ 1P ∈ P |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 {cab 2801 class class class wbr 5068 Qcnq 10276 1Qc1q 10277 <Q cltq 10282 Pcnp 10283 1Pc1p 10284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-ni 10296 df-pli 10297 df-mi 10298 df-lti 10299 df-plpq 10332 df-mpq 10333 df-ltpq 10334 df-enq 10335 df-nq 10336 df-erq 10337 df-plq 10338 df-mq 10339 df-1nq 10340 df-rq 10341 df-ltnq 10342 df-np 10405 df-1p 10406 |
This theorem is referenced by: 1idpr 10453 gt0srpr 10502 0r 10504 1sr 10505 m1r 10506 m1p1sr 10516 m1m1sr 10517 0lt1sr 10519 0idsr 10521 1idsr 10522 00sr 10523 recexsrlem 10527 mappsrpr 10532 ltpsrpr 10533 map2psrpr 10534 supsrlem 10535 |
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