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Theorem enqeq 10337

Description: Corollary of nqereu 10332: 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 1144	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
2		elpqn 10328	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
3	2	3ad2ant2 1130	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ∈ (N × N))
4		nqereu 10332	. . . 4 ⊢ (𝐵 ∈ (N × N) → ∃!𝑥 ∈ Q 𝑥 ~_Q 𝐵)
5		reurmo 3420	. . . 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 3141	. . 3 ⊢ (∃𝑥 ∈ Q 𝑥 ~_Q 𝐵 ↔ ∃𝑥(𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵))
8	6, 7	sylib 220	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → ∃*𝑥(𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵))
9		3simpb 1145	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐴 ∈ Q ∧ 𝐴 ~_Q 𝐵))
10		simp2 1133	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ∈ Q)
11		enqer 10324	. . . . 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 8290	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ~_Q 𝐵)
14	10, 13	jca 514	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐵 ∈ Q ∧ 𝐵 ~_Q 𝐵))
15		eleq1 2898	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Q ↔ 𝐴 ∈ Q))
16		breq1 5050	. . . 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 2898	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ Q ↔ 𝐵 ∈ Q))
19		breq1 5050	. . . 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 3695	. 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 1370	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐴 = 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∃*wmo 2620 ∃!wreu 3135 ∃*wrmo 3136 class class class wbr 5047 × cxp 5534 Er wer 8267 Ncnpi 10247 ~_Q ceq 10254 Qcnq 10255

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-oadd 8087 df-omul 8088 df-er 8270 df-ni 10275 df-mi 10277 df-lti 10278 df-enq 10314 df-nq 10315

This theorem is referenced by: nqereq 10338 ltsonq 10372

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