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Theorem axpre-lttri 10239
Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttri 10363. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 10263. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
axpre-lttri ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))

Proof of Theorem axpre-lttri
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 10205 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 10205 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 breq1 4812 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
4 eqeq1 2769 . . . . 5 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝐴 = ⟨𝑦, 0R⟩))
5 breq2 4813 . . . . 5 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ ⟨𝑦, 0R⟩ < 𝐴))
64, 5orbi12d 942 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)))
76notbid 309 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → (¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ ¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)))
83, 7bibi12d 336 . 2 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ ¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩)) ↔ (𝐴 <𝑦, 0R⟩ ↔ ¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴))))
9 breq2 4813 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
10 eqeq2 2776 . . . . 5 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 = ⟨𝑦, 0R⟩ ↔ 𝐴 = 𝐵))
11 breq1 4812 . . . . 5 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ < 𝐴𝐵 < 𝐴))
1210, 11orbi12d 942 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) ↔ (𝐴 = 𝐵𝐵 < 𝐴)))
1312notbid 309 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → (¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
149, 13bibi12d 336 . 2 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ↔ ¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)) ↔ (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴))))
15 ltsosr 10168 . . . 4 <R Or R
16 sotric 5224 . . . 4 (( <R Or R ∧ (𝑥R𝑦R)) → (𝑥 <R 𝑦 ↔ ¬ (𝑥 = 𝑦𝑦 <R 𝑥)))
1715, 16mpan 681 . . 3 ((𝑥R𝑦R) → (𝑥 <R 𝑦 ↔ ¬ (𝑥 = 𝑦𝑦 <R 𝑥)))
18 ltresr 10214 . . 3 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
19 vex 3353 . . . . . 6 𝑥 ∈ V
2019eqresr 10211 . . . . 5 (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝑥 = 𝑦)
21 ltresr 10214 . . . . 5 (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ 𝑦 <R 𝑥)
2220, 21orbi12i 938 . . . 4 ((⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝑥 = 𝑦𝑦 <R 𝑥))
2322notbii 311 . . 3 (¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ ¬ (𝑥 = 𝑦𝑦 <R 𝑥))
2417, 18, 233bitr4g 305 . 2 ((𝑥R𝑦R) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ ¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩)))
251, 2, 8, 14, 242gencl 3389 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  cop 4340   class class class wbr 4809   Or wor 5197  Rcnr 9940  0Rc0r 9941   <R cltr 9946  cr 10188   < cltrr 10193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-omul 7769  df-er 7947  df-ec 7949  df-qs 7953  df-ni 9947  df-pli 9948  df-mi 9949  df-lti 9950  df-plpq 9983  df-mpq 9984  df-ltpq 9985  df-enq 9986  df-nq 9987  df-erq 9988  df-plq 9989  df-mq 9990  df-1nq 9991  df-rq 9992  df-ltnq 9993  df-np 10056  df-1p 10057  df-plp 10058  df-ltp 10060  df-enr 10130  df-nr 10131  df-ltr 10134  df-0r 10135  df-r 10199  df-lt 10202
This theorem is referenced by: (None)
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