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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 7303 | . . . . 5 ⊢ P ⊆ (𝒫 Q × 𝒫 Q) | |
2 | 1 | sseli 3098 | . . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ (𝒫 Q × 𝒫 Q)) |
3 | 1st2nd2 6081 | . . . 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 3098 | . . . 4 ⊢ (𝐵 ∈ P → 𝐵 ∈ (𝒫 Q × 𝒫 Q)) |
6 | 1st2nd2 6081 | . . . 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 2156 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈(1st ‘𝐵), (2nd ‘𝐵)〉)) |
9 | xp1st 6071 | . . . . 5 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1st ‘𝐴) ∈ 𝒫 Q) | |
10 | 2, 9 | syl 14 | . . . 4 ⊢ (𝐴 ∈ P → (1st ‘𝐴) ∈ 𝒫 Q) |
11 | xp2nd 6072 | . . . . 5 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (2nd ‘𝐴) ∈ 𝒫 Q) | |
12 | 2, 11 | syl 14 | . . . 4 ⊢ (𝐴 ∈ P → (2nd ‘𝐴) ∈ 𝒫 Q) |
13 | opthg 4168 | . . . 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 409 | . . 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 1332 ∈ wcel 1481 𝒫 cpw 3515 〈cop 3535 × cxp 4545 ‘cfv 5131 1st c1st 6044 2nd c2nd 6045 Qcnq 7112 Pcnp 7123 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fv 5139 df-1st 6046 df-2nd 6047 df-inp 7298 |
This theorem is referenced by: genpassg 7358 addnqpr 7393 mulnqpr 7409 distrprg 7420 1idpr 7424 ltexpri 7445 addcanprg 7448 recexprlemex 7469 aptipr 7473 |
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