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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 11025 | . 2 ⊢ 1P = {𝑥 ∣ 𝑥 <Q 1Q} | |
2 | 1nq 10971 | . . 3 ⊢ 1Q ∈ Q | |
3 | nqpr 11057 | . . 3 ⊢ (1Q ∈ Q → {𝑥 ∣ 𝑥 <Q 1Q} ∈ P) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ {𝑥 ∣ 𝑥 <Q 1Q} ∈ P |
5 | 1, 4 | eqeltri 2822 | 1 ⊢ 1P ∈ P |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 {cab 2703 class class class wbr 5153 Qcnq 10895 1Qc1q 10896 <Q cltq 10901 Pcnp 10902 1Pc1p 10903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-oadd 8500 df-omul 8501 df-er 8734 df-ni 10915 df-pli 10916 df-mi 10917 df-lti 10918 df-plpq 10951 df-mpq 10952 df-ltpq 10953 df-enq 10954 df-nq 10955 df-erq 10956 df-plq 10957 df-mq 10958 df-1nq 10959 df-rq 10960 df-ltnq 10961 df-np 11024 df-1p 11025 |
This theorem is referenced by: 1idpr 11072 gt0srpr 11121 0r 11123 1sr 11124 m1r 11125 m1p1sr 11135 m1m1sr 11136 0lt1sr 11138 0idsr 11140 1idsr 11141 00sr 11142 recexsrlem 11146 mappsrpr 11151 ltpsrpr 11152 map2psrpr 11153 supsrlem 11154 |
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