preqlu - Intuitionistic Logic Explorer

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Theorem preqlu 6570

Description: Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.)

**Assertion**
Ref	Expression
preqlu	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵))))

**Proof of Theorem preqlu**
Step	Hyp	Ref	Expression
1		npsspw 6569	. . . . 5 ⊢ P ⊆ (𝒫 Q × 𝒫 Q)
2	1	sseli 2941	. . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ (𝒫 Q × 𝒫 Q))
3		1st2nd2 5801	. . . 4 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
4	2, 3	syl 14	. . 3 ⊢ (𝐴 ∈ P → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
5	1	sseli 2941	. . . 4 ⊢ (𝐵 ∈ P → 𝐵 ∈ (𝒫 Q × 𝒫 Q))
6		1st2nd2 5801	. . . 4 ⊢ (𝐵 ∈ (𝒫 Q × 𝒫 Q) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
7	5, 6	syl 14	. . 3 ⊢ (𝐵 ∈ P → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
8	4, 7	eqeqan12d 2055	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
9		xp1st 5792	. . . . 5 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (1^st ‘𝐴) ∈ 𝒫 Q)
10	2, 9	syl 14	. . . 4 ⊢ (𝐴 ∈ P → (1^st ‘𝐴) ∈ 𝒫 Q)
11		xp2nd 5793	. . . . 5 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → (2^nd ‘𝐴) ∈ 𝒫 Q)
12	2, 11	syl 14	. . . 4 ⊢ (𝐴 ∈ P → (2^nd ‘𝐴) ∈ 𝒫 Q)
13		opthg 3975	. . . 4 ⊢ (((1^st ‘𝐴) ∈ 𝒫 Q ∧ (2^nd ‘𝐴) ∈ 𝒫 Q) → (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ↔ ((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵))))
14	10, 12, 13	syl2anc 391	. . 3 ⊢ (𝐴 ∈ P → (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ↔ ((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵))))
15	14	adantr 261	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ↔ ((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵))))
16	8, 15	bitrd 177	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ((1^st ‘𝐴) = (1^st ‘𝐵) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 𝒫 cpw 3359 ⟨cop 3378 × cxp 4343 ‘cfv 4902 1^st c1st 5765 2^nd c2nd 5766 Qcnq 6378 Pcnp 6389

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fv 4910 df-1st 5767 df-2nd 5768 df-inp 6564

This theorem is referenced by: genpassg 6624 addnqpr 6659 mulnqpr 6675 distrprg 6686 1idpr 6690 ltexpri 6711 addcanprg 6714 recexprlemex 6735 aptipr 6739

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