preqlu - Intuitionistic Logic Explorer

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Theorem preqlu 7434

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 7433	. . . . 5 ⊢ P ⊆ (𝒫 Q × 𝒫 Q)
2	1	sseli 3143	. . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ (𝒫 Q × 𝒫 Q))
3		1st2nd2 6154	. . . 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 3143	. . . 4 ⊢ (𝐵 ∈ P → 𝐵 ∈ (𝒫 Q × 𝒫 Q))
6		1st2nd2 6154	. . . 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 2186	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
9		xp1st 6144	. . . . 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 6145	. . . . 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 4223	. . . 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 409	. . 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 274	. 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 187	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 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 𝒫 cpw 3566 ⟨cop 3586 × cxp 4609 ‘cfv 5198 1^st c1st 6117 2^nd c2nd 6118 Qcnq 7242 Pcnp 7253

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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fv 5206 df-1st 6119 df-2nd 6120 df-inp 7428

This theorem is referenced by: genpassg 7488 addnqpr 7523 mulnqpr 7539 distrprg 7550 1idpr 7554 ltexpri 7575 addcanprg 7578 recexprlemex 7599 aptipr 7603

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