nnnq - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > nnnq		GIF version

Theorem nnnq 7783

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 7676	. . . 4 ⊢ 1_o ∈ N
2		opelxpi 4804	. . . 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 7721	. . . 4 ⊢ ~_Q ∈ V
5	4	ecelqsi 6857	. . 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 7709	. 2 ⊢ Q = ((N × N) / ~_Q )
8	6, 7	eleqtrrdi 2332	1 ⊢ (𝐴 ∈ N → [⟨𝐴, 1_o⟩] ~_Q ∈ Q)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 2209 ⟨cop 3711 × cxp 4770 1_oc1o 6674 [cec 6799 / cqs 6800 Ncnpi 7633 ~_Q ceq 7640 Qcnq 7641

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-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 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-iinf 4733

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-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 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-br 4129 df-opab 4191 df-suc 4514 df-iom 4736 df-xp 4778 df-cnv 4780 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-1o 6681 df-ec 6803 df-qs 6807 df-ni 7665 df-enq 7708 df-nqqs 7709