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| Description: A positive fraction not in a positive real is an upper bound. Remark (1) of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| prub | ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ Q) → (¬ 𝐶 ∈ 𝐴 → 𝐵 <Q 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eleq1 2828 | . . . . . . 7 ⊢ (𝐵 = 𝐶 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
| 2 | 1 | biimpcd 249 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (𝐵 = 𝐶 → 𝐶 ∈ 𝐴)) | 
| 3 | 2 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → (𝐵 = 𝐶 → 𝐶 ∈ 𝐴)) | 
| 4 | prcdnq 11034 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → (𝐶 <Q 𝐵 → 𝐶 ∈ 𝐴)) | |
| 5 | 3, 4 | jaod 859 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → ((𝐵 = 𝐶 ∨ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴)) | 
| 6 | 5 | con3d 152 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → (¬ 𝐶 ∈ 𝐴 → ¬ (𝐵 = 𝐶 ∨ 𝐶 <Q 𝐵))) | 
| 7 | 6 | adantr 480 | . 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ Q) → (¬ 𝐶 ∈ 𝐴 → ¬ (𝐵 = 𝐶 ∨ 𝐶 <Q 𝐵))) | 
| 8 | elprnq 11032 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ Q) | |
| 9 | ltsonq 11010 | . . . 4 ⊢ <Q Or Q | |
| 10 | sotric 5621 | . . . 4 ⊢ (( <Q Or Q ∧ (𝐵 ∈ Q ∧ 𝐶 ∈ Q)) → (𝐵 <Q 𝐶 ↔ ¬ (𝐵 = 𝐶 ∨ 𝐶 <Q 𝐵))) | |
| 11 | 9, 10 | mpan 690 | . . 3 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐵 <Q 𝐶 ↔ ¬ (𝐵 = 𝐶 ∨ 𝐶 <Q 𝐵))) | 
| 12 | 8, 11 | sylan 580 | . 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ Q) → (𝐵 <Q 𝐶 ↔ ¬ (𝐵 = 𝐶 ∨ 𝐶 <Q 𝐵))) | 
| 13 | 7, 12 | sylibrd 259 | 1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 ∈ Q) → (¬ 𝐶 ∈ 𝐴 → 𝐵 <Q 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 Or wor 5590 Qcnq 10893 <Q cltq 10899 Pcnp 10900 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-oadd 8511 df-omul 8512 df-er 8746 df-ni 10913 df-mi 10915 df-lti 10916 df-ltpq 10951 df-enq 10952 df-nq 10953 df-ltnq 10959 df-np 11022 | 
| This theorem is referenced by: genpnnp 11046 psslinpr 11072 ltexprlem6 11082 ltexprlem7 11083 prlem936 11088 reclem4pr 11091 | 
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