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Theorem axpre-lttrn 6728
Description: Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 6757. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
axpre-lttrn ((A B 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))

Proof of Theorem axpre-lttrn
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 6687 . 2 (A ℝ ↔ x Rx, 0R⟩ = A)
2 elreal 6687 . 2 (B ℝ ↔ y Ry, 0R⟩ = B)
3 elreal 6687 . 2 (𝐶 ℝ ↔ z Rz, 0R⟩ = 𝐶)
4 breq1 3758 . . . 4 (⟨x, 0R⟩ = A → (⟨x, 0R⟩ <y, 0R⟩ ↔ A <y, 0R⟩))
54anbi1d 438 . . 3 (⟨x, 0R⟩ = A → ((⟨x, 0R⟩ <y, 0Ry, 0R⟩ <z, 0R⟩) ↔ (A <y, 0Ry, 0R⟩ <z, 0R⟩)))
6 breq1 3758 . . 3 (⟨x, 0R⟩ = A → (⟨x, 0R⟩ <z, 0R⟩ ↔ A <z, 0R⟩))
75, 6imbi12d 223 . 2 (⟨x, 0R⟩ = A → (((⟨x, 0R⟩ <y, 0Ry, 0R⟩ <z, 0R⟩) → ⟨x, 0R⟩ <z, 0R⟩) ↔ ((A <y, 0Ry, 0R⟩ <z, 0R⟩) → A <z, 0R⟩)))
8 breq2 3759 . . . 4 (⟨y, 0R⟩ = B → (A <y, 0R⟩ ↔ A < B))
9 breq1 3758 . . . 4 (⟨y, 0R⟩ = B → (⟨y, 0R⟩ <z, 0R⟩ ↔ B <z, 0R⟩))
108, 9anbi12d 442 . . 3 (⟨y, 0R⟩ = B → ((A <y, 0Ry, 0R⟩ <z, 0R⟩) ↔ (A < B B <z, 0R⟩)))
1110imbi1d 220 . 2 (⟨y, 0R⟩ = B → (((A <y, 0Ry, 0R⟩ <z, 0R⟩) → A <z, 0R⟩) ↔ ((A < B B <z, 0R⟩) → A <z, 0R⟩)))
12 breq2 3759 . . . 4 (⟨z, 0R⟩ = 𝐶 → (B <z, 0R⟩ ↔ B < 𝐶))
1312anbi2d 437 . . 3 (⟨z, 0R⟩ = 𝐶 → ((A < B B <z, 0R⟩) ↔ (A < B B < 𝐶)))
14 breq2 3759 . . 3 (⟨z, 0R⟩ = 𝐶 → (A <z, 0R⟩ ↔ A < 𝐶))
1513, 14imbi12d 223 . 2 (⟨z, 0R⟩ = 𝐶 → (((A < B B <z, 0R⟩) → A <z, 0R⟩) ↔ ((A < B B < 𝐶) → A < 𝐶)))
16 ltresr 6696 . . . . 5 (⟨x, 0R⟩ <y, 0R⟩ ↔ x <R y)
17 ltresr 6696 . . . . 5 (⟨y, 0R⟩ <z, 0R⟩ ↔ y <R z)
18 ltsosr 6652 . . . . . 6 <R Or R
19 ltrelsr 6626 . . . . . 6 <R ⊆ (R × R)
2018, 19sotri 4663 . . . . 5 ((x <R y y <R z) → x <R z)
2116, 17, 20syl2anb 275 . . . 4 ((⟨x, 0R⟩ <y, 0Ry, 0R⟩ <z, 0R⟩) → x <R z)
22 ltresr 6696 . . . 4 (⟨x, 0R⟩ <z, 0R⟩ ↔ x <R z)
2321, 22sylibr 137 . . 3 ((⟨x, 0R⟩ <y, 0Ry, 0R⟩ <z, 0R⟩) → ⟨x, 0R⟩ <z, 0R⟩)
2423a1i 9 . 2 ((x R y R z R) → ((⟨x, 0R⟩ <y, 0Ry, 0R⟩ <z, 0R⟩) → ⟨x, 0R⟩ <z, 0R⟩))
251, 2, 3, 7, 11, 15, 243gencl 2582 1 ((A B 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755  Rcnr 6281  0Rc0r 6282   <R cltr 6287  cr 6670   < cltrr 6675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-i1p 6449  df-iplp 6450  df-iltp 6452  df-enr 6614  df-nr 6615  df-ltr 6618  df-0r 6619  df-r 6681  df-lt 6684
This theorem is referenced by: (None)
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