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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 7389 | . . 3 ⊢ 1P = 〈{𝑦 ∣ 𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}〉 | |
2 | 1 | fveq2i 5473 | . 2 ⊢ (2nd ‘1P) = (2nd ‘〈{𝑦 ∣ 𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}〉) |
3 | ltnqex 7471 | . . 3 ⊢ {𝑦 ∣ 𝑦 <Q 1Q} ∈ V | |
4 | gtnqex 7472 | . . 3 ⊢ {𝑥 ∣ 1Q <Q 𝑥} ∈ V | |
5 | 3, 4 | op2nd 6097 | . 2 ⊢ (2nd ‘〈{𝑦 ∣ 𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}〉) = {𝑥 ∣ 1Q <Q 𝑥} |
6 | 2, 5 | eqtri 2178 | 1 ⊢ (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥} |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 {cab 2143 〈cop 3564 class class class wbr 3967 ‘cfv 5172 2nd c2nd 6089 1Qc1q 7203 <Q cltq 7207 1Pc1p 7214 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4081 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-iinf 4549 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-id 4255 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-2nd 6091 df-qs 6488 df-ni 7226 df-nqqs 7270 df-ltnqqs 7275 df-i1p 7389 |
This theorem is referenced by: 1idpru 7513 recexprlem1ssu 7556 recexprlemss1u 7558 |
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