preqlu - Intuitionistic Logic Explorer

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Theorem preqlu 7413

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 7412	. . . . 5 ⊢ P ⊆ (𝒫 Q × 𝒫 Q)
2	1	sseli 3138	. . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ (𝒫 Q × 𝒫 Q))
3		1st2nd2 6143	. . . 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 3138	. . . 4 ⊢ (𝐵 ∈ P → 𝐵 ∈ (𝒫 Q × 𝒫 Q))
6		1st2nd2 6143	. . . 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 2181	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
9		xp1st 6133	. . . . 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 6134	. . . . 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 4216	. . . 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 1343 ∈ wcel 2136 𝒫 cpw 3559 ⟨cop 3579 × cxp 4602 ‘cfv 5188 1^st c1st 6106 2^nd c2nd 6107 Qcnq 7221 Pcnp 7232

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-v 2728 df-sbc 2952 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-iota 5153 df-fun 5190 df-fv 5196 df-1st 6108 df-2nd 6109 df-inp 7407

This theorem is referenced by: genpassg 7467 addnqpr 7502 mulnqpr 7518 distrprg 7529 1idpr 7533 ltexpri 7554 addcanprg 7557 recexprlemex 7578 aptipr 7582

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