et-ltneverrefl - Mathbox for Ender Ting

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Theorem et-ltneverrefl 47287

Description: Less-than class is never reflexive. (Contributed by Ender Ting, 22-Nov-2024.) Prefer to specify theorem domain and then apply ltnri 11244. (New usage is discouraged.)

**Assertion**
Ref	Expression
et-ltneverrefl	⊢ ¬ 𝐴 < 𝐴

**Proof of Theorem et-ltneverrefl**
Step	Hyp	Ref	Expression
1		xrltnr 13059	. 2 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < 𝐴)
2		opelxp1 5662	. . . . 5 ⊢ (⟨𝐴, 𝐴⟩ ∈ (ℝ^* × ℝ^) → 𝐴 ∈ ℝ^)
3	2	con3i 154	. . . 4 ⊢ (¬ 𝐴 ∈ ℝ^* → ¬ ⟨𝐴, 𝐴⟩ ∈ (ℝ^* × ℝ^*))
4		ltrelxr 11195	. . . . 5 ⊢ < ⊆ (ℝ^* × ℝ^*)
5	4	sseli 3913	. . . 4 ⊢ (⟨𝐴, 𝐴⟩ ∈ < → ⟨𝐴, 𝐴⟩ ∈ (ℝ^* × ℝ^*))
6	3, 5	nsyl 140	. . 3 ⊢ (¬ 𝐴 ∈ ℝ^* → ¬ ⟨𝐴, 𝐴⟩ ∈ < )
7		df-br 5075	. . 3 ⊢ (𝐴 < 𝐴 ↔ ⟨𝐴, 𝐴⟩ ∈ < )
8	6, 7	sylnibr 329	. 2 ⊢ (¬ 𝐴 ∈ ℝ^* → ¬ 𝐴 < 𝐴)
9	1, 8	pm2.61i 182	1 ⊢ ¬ 𝐴 < 𝐴

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∈ wcel 2114 ⟨cop 4563 class class class wbr 5074 × cxp 5618 ℝ^*cxr 11167 < clt 11168

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 2184 ax-ext 2707 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-cnex 11083 ax-resscn 11084 ax-pre-lttri 11101 ax-pre-lttrn 11102

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-br 5075 df-opab 5137 df-mpt 5156 df-id 5515 df-po 5528 df-so 5529 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-er 8632 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173

This theorem is referenced by: (None)

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