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Mirrors > Home > ILE Home > Th. List > 1prl | GIF version |
Description: The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.) |
Ref | Expression |
---|---|
1prl | ⊢ (1st ‘1P) = {𝑥 ∣ 𝑥 <Q 1Q} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-i1p 7444 | . . 3 ⊢ 1P = 〈{𝑥 ∣ 𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}〉 | |
2 | 1 | fveq2i 5513 | . 2 ⊢ (1st ‘1P) = (1st ‘〈{𝑥 ∣ 𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}〉) |
3 | ltnqex 7526 | . . 3 ⊢ {𝑥 ∣ 𝑥 <Q 1Q} ∈ V | |
4 | gtnqex 7527 | . . 3 ⊢ {𝑦 ∣ 1Q <Q 𝑦} ∈ V | |
5 | 3, 4 | op1st 6140 | . 2 ⊢ (1st ‘〈{𝑥 ∣ 𝑥 <Q 1Q}, {𝑦 ∣ 1Q <Q 𝑦}〉) = {𝑥 ∣ 𝑥 <Q 1Q} |
6 | 2, 5 | eqtri 2198 | 1 ⊢ (1st ‘1P) = {𝑥 ∣ 𝑥 <Q 1Q} |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 {cab 2163 〈cop 3594 class class class wbr 4000 ‘cfv 5211 1st c1st 6132 1Qc1q 7258 <Q cltq 7262 1Pc1p 7269 |
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 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-iinf 4583 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-1st 6134 df-qs 6534 df-ni 7281 df-nqqs 7325 df-ltnqqs 7330 df-i1p 7444 |
This theorem is referenced by: 1idprl 7567 recexprlem1ssl 7610 recexprlemss1l 7612 |
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