preqlu - Intuitionistic Logic Explorer

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Theorem preqlu 7833

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 7832	. . . . 5 ⊢ P ⊆ (𝒫 Q × 𝒫 Q)
2	1	sseli 3244	. . . 4 ⊢ (𝐴 ∈ P → 𝐴 ∈ (𝒫 Q × 𝒫 Q))
3		1st2nd2 6403	. . . 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 3244	. . . 4 ⊢ (𝐵 ∈ P → 𝐵 ∈ (𝒫 Q × 𝒫 Q))
6		1st2nd2 6403	. . . 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 2254	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩))
9		xp1st 6393	. . . . 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 6394	. . . . 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 4376	. . . 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 415	. . 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 276	. 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 188	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 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 𝒫 cpw 3688 ⟨cop 3711 × cxp 4770 ‘cfv 5375 1^st c1st 6366 2^nd c2nd 6367 Qcnq 7641 Pcnp 7652

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4247 ax-pow 4309 ax-pr 4344 ax-un 4576

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-v 2823 df-sbc 3052 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-br 4129 df-opab 4191 df-mpt 4192 df-id 4436 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-iota 5335 df-fun 5377 df-fv 5383 df-1st 6368 df-2nd 6369 df-inp 7827

This theorem is referenced by: genpassg 7887 addnqpr 7922 mulnqpr 7938 distrprg 7949 1idpr 7953 ltexpri 7974 addcanprg 7977 recexprlemex 7998 aptipr 8002

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