xrleloe - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xrleloe		Structured version Visualization version GIF version

Theorem xrleloe 13040

Description: 'Less than or equal' expressed in terms of 'less than' or 'equals', for extended reals. (Contributed by NM, 19-Jan-2006.)

**Assertion**
Ref	Expression
xrleloe	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))

**Proof of Theorem xrleloe**
Step	Hyp	Ref	Expression
1		xrlenlt 11174	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
2		xrlttri 13035	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴 < 𝐵)))
3	2	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴 < 𝐵)))
4	3	con2bid 354	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ ¬ 𝐵 < 𝐴))
5		eqcom 2738	. . . . 5 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
6	5	orbi1i 913	. . . 4 ⊢ ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ (𝐴 = 𝐵 ∨ 𝐴 < 𝐵))
7		orcom 870	. . . 4 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))
8	6, 7	bitri 275	. . 3 ⊢ ((𝐵 = 𝐴 ∨ 𝐴 < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))
9	4, 8	bitr3di 286	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (¬ 𝐵 < 𝐴 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))
10	1, 9	bitrd 279	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2111 class class class wbr 5091 ℝ^*cxr 11142 < clt 11143 ≤ cle 11144

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-pre-lttri 11077 ax-pre-lttrn 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-po 5524 df-so 5525 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149

This theorem is referenced by: xrleltne 13041 dfle2 13043 xrltle 13045 xrleid 13047 xrlelttr 13052 xrltletr 13053 xrletr 13054 nltpnft 13060 ngtmnft 13062 xmulge0 13180 xlemul1a 13184 xadddi2 13193 prunioo 13378 xrsxmet 24723 metds0 24764 metdseq0 24768 metnrmlem1a 24772 icombl 25490 ioombl 25491 volivth 25533 vitalilem4 25537 itg2gt0 25686 deg1sublt 26040 xrge0addgt0 32993 xrge0adddir 32994 icorempo 37384 icceuelpartlem 47465