Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > preqlu | GIF version |
Description: Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Ref | Expression |
---|---|
preqlu | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | npsspw 7247 | . . . . 5 ⊢ P ⊆ (𝒫 Q × 𝒫 Q) | |
2 | 1 | sseli 3063 | . . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ (𝒫 Q × 𝒫 Q)) |
3 | 1st2nd2 6041 | . . . 4 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝐴 ∈ P → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
5 | 1 | sseli 3063 | . . . 4 ⊢ (𝐵 ∈ P → 𝐵 ∈ (𝒫 Q × 𝒫 Q)) |
6 | 1st2nd2 6041 | . . . 4 ⊢ (𝐵 ∈ (𝒫 Q × 𝒫 Q) → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ (𝐵 ∈ P → 𝐵 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉) |
8 | 4, 7 | eqeqan12d 2133 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉)) |
9 | xp1st 6031 | . . . . 5 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1st ‘𝐴) ∈ 𝒫 Q) | |
10 | 2, 9 | syl 14 | . . . 4 ⊢ (𝐴 ∈ P → (1st ‘𝐴) ∈ 𝒫 Q) |
11 | xp2nd 6032 | . . . . 5 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (2nd ‘𝐴) ∈ 𝒫 Q) | |
12 | 2, 11 | syl 14 | . . . 4 ⊢ (𝐴 ∈ P → (2nd ‘𝐴) ∈ 𝒫 Q) |
13 | opthg 4130 | . . . 4 ⊢ (((1st ‘𝐴) ∈ 𝒫 Q ∧ (2nd ‘𝐴) ∈ 𝒫 Q) → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)))) | |
14 | 10, 12, 13 | syl2anc 408 | . . 3 ⊢ (𝐴 ∈ P → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)))) |
15 | 14 | adantr 274 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)))) |
16 | 8, 15 | bitrd 187 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 ∈ wcel 1465 𝒫 cpw 3480 〈cop 3500 × cxp 4507 ‘cfv 5093 1st c1st 6004 2nd c2nd 6005 Qcnq 7056 Pcnp 7067 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fv 5101 df-1st 6006 df-2nd 6007 df-inp 7242 |
This theorem is referenced by: genpassg 7302 addnqpr 7337 mulnqpr 7353 distrprg 7364 1idpr 7368 ltexpri 7389 addcanprg 7392 recexprlemex 7413 aptipr 7417 |
Copyright terms: Public domain | W3C validator |