et-ltneverrefl - Mathbox for Ender Ting

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Theorem et-ltneverrefl 47803

Description: Less-than class is never reflexive. (Contributed by Ender Ting, 22-Nov-2024.) Prefer to specify theorem domain and then apply ltnri 11390. (New usage is discouraged.)

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

**Proof of Theorem et-ltneverrefl**
Step	Hyp	Ref	Expression
1		xrltnr 13217	. 2 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < 𝐴)
2		opelxp1 5689	. . . . 5 ⊢ (⟨𝐴, 𝐴⟩ ∈ (ℝ^* × ℝ^) → 𝐴 ∈ ℝ^)
3	2	con3i 155	. . . 4 ⊢ (¬ 𝐴 ∈ ℝ^* → ¬ ⟨𝐴, 𝐴⟩ ∈ (ℝ^* × ℝ^*))
4		ltrelxr 11341	. . . . 5 ⊢ < ⊆ (ℝ^* × ℝ^*)
5	4	sseli 3926	. . . 4 ⊢ (⟨𝐴, 𝐴⟩ ∈ < → ⟨𝐴, 𝐴⟩ ∈ (ℝ^* × ℝ^*))
6	3, 5	nsyl 141	. . 3 ⊢ (¬ 𝐴 ∈ ℝ^* → ¬ ⟨𝐴, 𝐴⟩ ∈ < )
7		df-br 5103	. . 3 ⊢ (𝐴 < 𝐴 ↔ ⟨𝐴, 𝐴⟩ ∈ < )
8	6, 7	sylnibr 332	. 2 ⊢ (¬ 𝐴 ∈ ℝ^* → ¬ 𝐴 < 𝐴)
9	1, 8	pm2.61i 184	1 ⊢ ¬ 𝐴 < 𝐴

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ∈ wcel 2145 ⟨cop 4589 class class class wbr 5102 × cxp 5645 ℝ^*cxr 11313 < clt 11314

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11227 ax-resscn 11228 ax-pre-lttri 11245 ax-pre-lttrn 11246

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5542 df-po 5555 df-so 5556 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-er 8695 df-en 8952 df-dom 8953 df-sdom 8954 df-pnf 11316 df-mnf 11317 df-xr 11318 df-ltxr 11319

This theorem is used by: (None)

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