axlttri - Metamath Proof Explorer

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Theorem axlttri 5475

Description: Ordering on reals satisfies strict trichotomy. Axiom 22 of 27 for real and complex numbers, derived from ZF set theory. (This restates pre-axlttri 5259 with ordering on the extended reals.)

**Assertion**
Ref	Expression
axlttri	⊢ ((A ∈ ℝ ⋀ B ∈ ℝ) → (A < B ↔ ¬ (A = B ⋁ B < A)))

**Proof of Theorem axlttri**
Step	Hyp	Ref	Expression
1		pre-axlttri 5259	. 2 ⊢ ((A ∈ ℝ ⋀ B ∈ ℝ) → (A <_ℝ B ↔ ¬ (A = B ⋁ B <_ℝ A)))
2		ltxrltt 5472	. 2 ⊢ ((A ∈ ℝ ⋀ B ∈ ℝ) → (A < B ↔ A <_ℝ B))
3		ltxrltt 5472	. . . . 5 ⊢ ((B ∈ ℝ ⋀ A ∈ ℝ) → (B < A ↔ B <_ℝ A))
4	3	ancoms 436	. . . 4 ⊢ ((A ∈ ℝ ⋀ B ∈ ℝ) → (B < A ↔ B <_ℝ A))
5	4	orbi2d 612	. . 3 ⊢ ((A ∈ ℝ ⋀ B ∈ ℝ) → ((A = B ⋁ B < A) ↔ (A = B ⋁ B <_ℝ A)))
6	5	negbid 609	. 2 ⊢ ((A ∈ ℝ ⋀ B ∈ ℝ) → (¬ (A = B ⋁ B < A) ↔ ¬ (A = B ⋁ B <_ℝ A)))
7	1, 2, 6	3bitr4d 548	1 ⊢ ((A ∈ ℝ ⋀ B ∈ ℝ) → (A < B ↔ ¬ (A = B ⋁ B < A)))

Colors of variables: wff set class

Syntax hints: ¬ wn 2 → wi 3 ↔ wb 146 ⋁ wo 222 ⋀ wa 223 = wceq 953 ∈ wcel 955 class class class wbr 2609 ℝcr 5205 <_ℝ cltrr 5210 < clt 5458

This theorem is referenced by: ltso 5484 lttri4t 5487 leloet 5491 ltnsymt 5505 xrlttrit 5525 recgt0i 5770 arch 6018 expordt 6533 sqrlem13 6615

This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 959 ax-gen 960 ax-8 961 ax-9 962 ax-10 963 ax-11 964 ax-12 965 ax-13 966 ax-14 967 ax-17 968 ax-4 970 ax-5o 972 ax-6o 975 ax-9o 1119 ax-10o 1136 ax-16 1206 ax-11o 1213 ax-ext 1452 ax-rep 2683 ax-sep 2693 ax-nul 2700 ax-pow 2732 ax-pr 2769 ax-un 2857 ax-inf2 4597

This theorem depends on definitions: df-bi 147 df-or 224 df-an 225 df-3or 774 df-3an 775 df-ex 978 df-sb 1168 df-eu 1375 df-mo 1376 df-clab 1457 df-cleq 1462 df-clel 1465 df-ne 1579 df-nel 1580 df-ral 1641 df-rex 1642 df-reu 1643 df-rab 1644 df-v 1803 df-sbc 1932 df-csb 1992 df-dif 2039 df-un 2040 df-in 2041 df-ss 2043 df-pss 2045 df-nul 2271 df-if 2352 df-pw 2392 df-sn 2402 df-pr 2403 df-tp 2405 df-op 2406 df-uni 2494 df-int 2524 df-iun 2558 df-br 2610 df-opab 2657 df-tr 2671 df-eprel 2821 df-id 2824 df-po 2831 df-so 2841 df-fr 2907 df-we 2924 df-ord 2941 df-on 2942 df-lim 2943 df-suc 2944 df-om 3122 df-xp 3174 df-rel 3175 df-cnv 3176 df-co 3177 df-dm 3178 df-rn 3179 df-res 3180 df-ima 3181 df-fun 3182 df-fn 3183 df-f 3184 df-f1 3185 df-fo 3186 df-f1o 3187 df-fv 3188 df-rdg 3917 df-opr 3950 df-oprab 3951 df-1st 4063 df-2nd 4064 df-1o 4117 df-oadd 4119 df-omul 4120 df-er 4245 df-ec 4247 df-qs 4250 df-en 4351 df-dom 4352 df-sdom 4353 df-ni 4972 df-pli 4973 df-mi 4974 df-lti 4975 df-plpq 5007 df-mpq 5008 df-enq 5009 df-nq 5010 df-plq 5011 df-mq 5012 df-rq 5013 df-ltq 5014 df-1q 5015 df-np 5058 df-1p 5059 df-plp 5060 df-ltp 5062 df-enr 5138 df-nr 5139 df-ltr 5142 df-0r 5143 df-c 5212 df-r 5216 df-lt 5219 df-pnf 5459 df-mnf 5460 df-xr 5461 df-ltxr 5462