nnnq - Intuitionistic Logic Explorer

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Theorem nnnq 7796

Description: The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)

**Assertion**
Ref	Expression
nnnq	⊢ (𝐴 ∈ N → [⟨𝐴, 1_o⟩] ~_Q ∈ Q)

**Proof of Theorem nnnq**
Step	Hyp	Ref	Expression
1		1pi 7689	. . . 4 ⊢ 1_o ∈ N
2		opelxpi 4808	. . . 4 ⊢ ((𝐴 ∈ N ∧ 1_o ∈ N) → ⟨𝐴, 1_o⟩ ∈ (N × N))
3	1, 2	mpan2 429	. . 3 ⊢ (𝐴 ∈ N → ⟨𝐴, 1_o⟩ ∈ (N × N))
4		enqex 7734	. . . 4 ⊢ ~_Q ∈ V
5	4	ecelqsi 6867	. . 3 ⊢ (⟨𝐴, 1_o⟩ ∈ (N × N) → [⟨𝐴, 1_o⟩] ~_Q ∈ ((N × N) / ~_Q ))
6	3, 5	syl 14	. 2 ⊢ (𝐴 ∈ N → [⟨𝐴, 1_o⟩] ~_Q ∈ ((N × N) / ~_Q ))
7		df-nqqs 7722	. 2 ⊢ Q = ((N × N) / ~_Q )
8	6, 7	eleqtrrdi 2332	1 ⊢ (𝐴 ∈ N → [⟨𝐴, 1_o⟩] ~_Q ∈ Q)

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∈ wcel 2209 ⟨cop 3712 × cxp 4774 1_oc1o 6684 [cec 6809 / cqs 6810 Ncnpi 7646 ~_Q ceq 7653 Qcnq 7654

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 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 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-iinf 4737

This proof 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-ne 2421 df-ral 2533 df-rex 2534 df-v 2823 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-br 4132 df-opab 4194 df-suc 4518 df-iom 4740 df-xp 4782 df-cnv 4784 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-1o 6691 df-ec 6813 df-qs 6817 df-ni 7678 df-enq 7721 df-nqqs 7722