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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 7465 | . . 3 ⊢ 1P = 〈{𝑦 ∣ 𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}〉 | |
2 | 1 | fveq2i 5518 | . 2 ⊢ (2nd ‘1P) = (2nd ‘〈{𝑦 ∣ 𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}〉) |
3 | ltnqex 7547 | . . 3 ⊢ {𝑦 ∣ 𝑦 <Q 1Q} ∈ V | |
4 | gtnqex 7548 | . . 3 ⊢ {𝑥 ∣ 1Q <Q 𝑥} ∈ V | |
5 | 3, 4 | op2nd 6147 | . 2 ⊢ (2nd ‘〈{𝑦 ∣ 𝑦 <Q 1Q}, {𝑥 ∣ 1Q <Q 𝑥}〉) = {𝑥 ∣ 1Q <Q 𝑥} |
6 | 2, 5 | eqtri 2198 | 1 ⊢ (2nd ‘1P) = {𝑥 ∣ 1Q <Q 𝑥} |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 {cab 2163 〈cop 3595 class class class wbr 4003 ‘cfv 5216 2nd c2nd 6139 1Qc1q 7279 <Q cltq 7283 1Pc1p 7290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-2nd 6141 df-qs 6540 df-ni 7302 df-nqqs 7346 df-ltnqqs 7351 df-i1p 7465 |
This theorem is referenced by: 1idpru 7589 recexprlem1ssu 7632 recexprlemss1u 7634 |
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