preqlu - Intuitionistic Logic Explorer

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Theorem preqlu 6978

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 6977	. . . . 5 ⊢ P ⊆ (𝒫 Q × 𝒫 Q)
2	1	sseli 3010	. . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ (𝒫 Q × 𝒫 Q))
3		1st2nd2 5904	. . . 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 3010	. . . 4 ⊢ (𝐵 ∈ P → 𝐵 ∈ (𝒫 Q × 𝒫 Q))
6		1st2nd2 5904	. . . 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 2100	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
9		xp1st 5895	. . . . 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 5896	. . . . 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 4041	. . . 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 403	. . 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 270	. 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 186	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 102 ↔ wb 103 = wceq 1287 ∈ wcel 1436 𝒫 cpw 3415 ⟨cop 3434 × cxp 4411 ‘cfv 4983 1^st c1st 5868 2^nd c2nd 5869 Qcnq 6786 Pcnp 6797

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3934 ax-pow 3986 ax-pr 4012 ax-un 4236

This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ral 2360 df-rex 2361 df-v 2617 df-sbc 2830 df-un 2992 df-in 2994 df-ss 3001 df-pw 3417 df-sn 3437 df-pr 3438 df-op 3440 df-uni 3639 df-br 3823 df-opab 3877 df-mpt 3878 df-id 4096 df-xp 4419 df-rel 4420 df-cnv 4421 df-co 4422 df-dm 4423 df-rn 4424 df-iota 4948 df-fun 4985 df-fv 4991 df-1st 5870 df-2nd 5871 df-inp 6972

This theorem is referenced by: genpassg 7032 addnqpr 7067 mulnqpr 7083 distrprg 7094 1idpr 7098 ltexpri 7119 addcanprg 7122 recexprlemex 7143 aptipr 7147

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