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Mirrors > Home > ILE Home > Th. List > 1pru | GIF version |
Description: The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.) |
Ref | Expression |
---|---|
1pru | ⊢ (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-i1p 7275 | . . 3 ⊢ 1P = 〈{𝑦 ∣ 𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}〉 | |
2 | 1 | fveq2i 5424 | . 2 ⊢ (2nd ‘1P) = (2nd ‘〈{𝑦 ∣ 𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}〉) |
3 | ltnqex 7357 | . . 3 ⊢ {𝑦 ∣ 𝑦 <Q 1Q} ∈ V | |
4 | gtnqex 7358 | . . 3 ⊢ {𝑥 ∣ 1Q <Q 𝑥} ∈ V | |
5 | 3, 4 | op2nd 6045 | . 2 ⊢ (2nd ‘〈{𝑦 ∣ 𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}〉) = {𝑥 ∣ 1Q <Q 𝑥} |
6 | 2, 5 | eqtri 2160 | 1 ⊢ (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥} |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 {cab 2125 〈cop 3530 class class class wbr 3929 ‘cfv 5123 2nd c2nd 6037 1Qc1q 7089 <Q cltq 7093 1Pc1p 7100 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-2nd 6039 df-qs 6435 df-ni 7112 df-nqqs 7156 df-ltnqqs 7161 df-i1p 7275 |
This theorem is referenced by: 1idpru 7399 recexprlem1ssu 7442 recexprlemss1u 7444 |
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