et-ltneverrefl - Mathbox for Ender Ting

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Theorem et-ltneverrefl 46869

Description: Less-than class is never reflexive. (Contributed by Ender Ting, 22-Nov-2024.) Prefer to specify theorem domain and then apply ltnri 11283. (New usage is discouraged.)

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

**Proof of Theorem et-ltneverrefl**
Step	Hyp	Ref	Expression
1		xrltnr 13079	. 2 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < 𝐴)
2		opelxp1 5680	. . . . 5 ⊢ (⟨𝐴, 𝐴⟩ ∈ (ℝ^* × ℝ^) → 𝐴 ∈ ℝ^)
3	2	con3i 154	. . . 4 ⊢ (¬ 𝐴 ∈ ℝ^* → ¬ ⟨𝐴, 𝐴⟩ ∈ (ℝ^* × ℝ^*))
4		ltrelxr 11235	. . . . 5 ⊢ < ⊆ (ℝ^* × ℝ^*)
5	4	sseli 3942	. . . 4 ⊢ (⟨𝐴, 𝐴⟩ ∈ < → ⟨𝐴, 𝐴⟩ ∈ (ℝ^* × ℝ^*))
6	3, 5	nsyl 140	. . 3 ⊢ (¬ 𝐴 ∈ ℝ^* → ¬ ⟨𝐴, 𝐴⟩ ∈ < )
7		df-br 5108	. . 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 2109 ⟨cop 4595 class class class wbr 5107 × cxp 5636 ℝ^*cxr 11207 < clt 11208

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-pre-lttri 11142 ax-pre-lttrn 11143

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213

This theorem is referenced by: tworepnotupword 46884

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