Theorem axpre-lttrn 7883

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 7925. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
axpre-lttrn	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))

Proof of Theorem axpre-lttrn

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elreal 7827	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 7827	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 7827	. 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		breq1 4007	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
5	4	anbi1d 465	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) ↔ (𝐴 <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩)))
6		breq1 4007	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑧, 0R⟩))
7	5, 6	imbi12d 234	. 2 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → ⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩) ↔ ((𝐴 <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → 𝐴 <ℝ ⟨𝑧, 0R⟩)))
8		breq2 4008	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
9		breq1 4007	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝐵 <ℝ ⟨𝑧, 0R⟩))
10	8, 9	anbi12d 473	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) ↔ (𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ ⟨𝑧, 0R⟩)))
11	10	imbi1d 231	. 2 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (((𝐴 <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → 𝐴 <ℝ ⟨𝑧, 0R⟩) ↔ ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ ⟨𝑧, 0R⟩) → 𝐴 <ℝ ⟨𝑧, 0R⟩)))
12		breq2 4008	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐵 <ℝ ⟨𝑧, 0R⟩ ↔ 𝐵 <ℝ 𝐶))
13	12	anbi2d 464	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ ⟨𝑧, 0R⟩) ↔ (𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶)))
14		breq2 4008	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ 𝐶))
15	13, 14	imbi12d 234	. 2 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ ⟨𝑧, 0R⟩) → 𝐴 <ℝ ⟨𝑧, 0R⟩) ↔ ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶)))
16		ltresr 7838	. . . . 5 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
17		ltresr 7838	. . . . 5 ⊢ (⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝑦 <R 𝑧)
18		ltsosr 7763	. . . . . 6 ⊢ <R Or R
19		ltrelsr 7737	. . . . . 6 ⊢ <R ⊆ (R × R)
20	18, 19	sotri 5025	. . . . 5 ⊢ ((𝑥 <R 𝑦 ∧ 𝑦 <R 𝑧) → 𝑥 <R 𝑧)
21	16, 17, 20	syl2anb 291	. . . 4 ⊢ ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → 𝑥 <R 𝑧)
22		ltresr 7838	. . . 4 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
23	21, 22	sylibr 134	. . 3 ⊢ ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → ⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩)
24	23	a1i 9	. 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → ⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩))
25	1, 2, 3, 7, 11, 15, 24	3gencl 2772	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ⟨cop 3596   class class class wbr 4004  Rcnr 7296  0_Rc0r 7297   <_R cltr 7302  ℝcr 7810   <_ℝ cltrr 7815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-i1p 7466  df-iplp 7467  df-iltp 7469  df-enr 7725  df-nr 7726  df-ltr 7729  df-0r 7730  df-r 7821  df-lt 7824

This theorem is referenced by: (None)