gtnqex - Intuitionistic Logic Explorer

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Theorem gtnqex 7918

Description: The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)

**Assertion**
Ref	Expression
gtnqex	⊢ {𝑥 ∣ 𝐴 <_Q 𝑥} ∈ V

**Proof of Theorem gtnqex**
Step	Hyp	Ref	Expression
1		nqex 7731	. 2 ⊢ Q ∈ V
2		ltrelnq 7733	. . . . 5 ⊢ <_Q ⊆ (Q × Q)
3	2	brel 4827	. . . 4 ⊢ (𝐴 <_Q 𝑥 → (𝐴 ∈ Q ∧ 𝑥 ∈ Q))
4	3	simprd 114	. . 3 ⊢ (𝐴 <_Q 𝑥 → 𝑥 ∈ Q)
5	4	abssi 3323	. 2 ⊢ {𝑥 ∣ 𝐴 <_Q 𝑥} ⊆ Q
6	1, 5	ssexi 4271	1 ⊢ {𝑥 ∣ 𝐴 <_Q 𝑥} ∈ V

Colors of variables: wff set class

This proof depends on syntax axioms: ∈ wcel 2209 {cab 2224 Vcvv 2821 class class class wbr 4130 Qcnq 7648 <_Q cltq 7653

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-coll 4246 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-iinf 4735

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-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-id 4438 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-qs 6813 df-ni 7672 df-nqqs 7716 df-ltnqqs 7721