Theorem axpre-lttrn 11080

Description: Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttrn 11209. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 11104. (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 11045	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 11045	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 11045	. 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		breq1 5089	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
5	4	anbi1d 632	. . 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 5089	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑧, 0R⟩))
7	5, 6	imbi12d 344	. 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 5090	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
9		breq1 5089	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝐵 <ℝ ⟨𝑧, 0R⟩))
10	8, 9	anbi12d 633	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) ↔ (𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ ⟨𝑧, 0R⟩)))
11	10	imbi1d 341	. 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 5090	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐵 <ℝ ⟨𝑧, 0R⟩ ↔ 𝐵 <ℝ 𝐶))
13	12	anbi2d 631	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ ⟨𝑧, 0R⟩) ↔ (𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶)))
14		breq2 5090	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ 𝐶))
15	13, 14	imbi12d 344	. 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 11054	. . . . 5 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
17		ltresr 11054	. . . . 5 ⊢ (⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝑦 <R 𝑧)
18		ltsosr 11008	. . . . . 6 ⊢ <R Or R
19		ltrelsr 10982	. . . . . 6 ⊢ <R ⊆ (R × R)
20	18, 19	sotri 6084	. . . . 5 ⊢ ((𝑥 <R 𝑦 ∧ 𝑦 <R 𝑧) → 𝑥 <R 𝑧)
21	16, 17, 20	syl2anb 599	. . . 4 ⊢ ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → 𝑥 <R 𝑧)
22		ltresr 11054	. . . 4 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
23	21, 22	sylibr 234	. . 3 ⊢ ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → ⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩)
24	23	a1i 11	. 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 3474	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ⟨cop 4574   class class class wbr 5086  Rcnr 10779  0_Rc0r 10780   <_R cltr 10785  ℝcr 11028   <_ℝ cltrr 11033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-oadd 8402  df-omul 8403  df-er 8636  df-ec 8638  df-qs 8642  df-ni 10786  df-pli 10787  df-mi 10788  df-lti 10789  df-plpq 10822  df-mpq 10823  df-ltpq 10824  df-enq 10825  df-nq 10826  df-erq 10827  df-plq 10828  df-mq 10829  df-1nq 10830  df-rq 10831  df-ltnq 10832  df-np 10895  df-1p 10896  df-plp 10897  df-ltp 10899  df-enr 10969  df-nr 10970  df-ltr 10973  df-0r 10974  df-r 11039  df-lt 11042

This theorem is referenced by: (None)