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Theorem enqeq 10828

Description: Corollary of nqereu 10823: if two fractions are both reduced and equivalent, then they are equal. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)

**Assertion**
Ref	Expression
enqeq	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐴 = 𝐵)

Proof of Theorem enqeq Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpa 1148	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
2		elpqn 10819	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
3	2	3ad2ant2 1134	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ∈ (N × N))
4		nqereu 10823	. . . 4 ⊢ (𝐵 ∈ (N × N) → ∃!𝑥 ∈ Q 𝑥 ~_Q 𝐵)
5		reurmo 3346	. . . 4 ⊢ (∃!𝑥 ∈ Q 𝑥 ~_Q 𝐵 → ∃*𝑥 ∈ Q 𝑥 ~_Q 𝐵)
6	3, 4, 5	3syl 18	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → ∃*𝑥 ∈ Q 𝑥 ~_Q 𝐵)
7		df-rmo 3343	. . 3 ⊢ (∃𝑥 ∈ Q 𝑥 ~_Q 𝐵 ↔ ∃𝑥(𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵))
8	6, 7	sylib 218	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → ∃*𝑥(𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵))
9		3simpb 1149	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐴 ∈ Q ∧ 𝐴 ~_Q 𝐵))
10		simp2 1137	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ∈ Q)
11		enqer 10815	. . . . 5 ⊢ ~_Q Er (N × N)
12	11	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → ~_Q Er (N × N))
13	12, 3	erref 8645	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ~_Q 𝐵)
14	10, 13	jca 511	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐵 ∈ Q ∧ 𝐵 ~_Q 𝐵))
15		eleq1 2816	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Q ↔ 𝐴 ∈ Q))
16		breq1 5095	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ~_Q 𝐵 ↔ 𝐴 ~_Q 𝐵))
17	15, 16	anbi12d 632	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵) ↔ (𝐴 ∈ Q ∧ 𝐴 ~_Q 𝐵)))
18		eleq1 2816	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ Q ↔ 𝐵 ∈ Q))
19		breq1 5095	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ~_Q 𝐵 ↔ 𝐵 ~_Q 𝐵))
20	18, 19	anbi12d 632	. . 3 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵) ↔ (𝐵 ∈ Q ∧ 𝐵 ~_Q 𝐵)))
21	17, 20	moi 3678	. 2 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ ∃*𝑥(𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵) ∧ ((𝐴 ∈ Q ∧ 𝐴 ~_Q 𝐵) ∧ (𝐵 ∈ Q ∧ 𝐵 ~_Q 𝐵))) → 𝐴 = 𝐵)
22	1, 8, 9, 14, 21	syl112anc 1376	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐴 = 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃*wmo 2531 ∃!wreu 3341 ∃*wrmo 3342 class class class wbr 5092 × cxp 5617 Er wer 8622 Ncnpi 10738 ~_Q ceq 10745 Qcnq 10746

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pr 5371 ax-un 7671

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-oadd 8392 df-omul 8393 df-er 8625 df-ni 10766 df-mi 10768 df-lti 10769 df-enq 10805 df-nq 10806

This theorem is referenced by: nqereq 10829 ltsonq 10863

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