et-ltneverrefl - Mathbox for Ender Ting

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Theorem et-ltneverrefl 48084

Description: Less-than class is never reflexive. (Contributed by Ender Ting, 22-Nov-2024.) Prefer to specify theorem domain and then apply ltnri 11443. (New usage is discouraged.)

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

**Proof of Theorem et-ltneverrefl**
Step	Hyp	Ref	Expression
1		xrltnr 13272	. 2 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < 𝐴)
2		opelxp1 5697	. . . . 5 ⊢ (⟨𝐴, 𝐴⟩ ∈ (ℝ^* × ℝ^) → 𝐴 ∈ ℝ^)
3	2	con3i 155	. . . 4 ⊢ (¬ 𝐴 ∈ ℝ^* → ¬ ⟨𝐴, 𝐴⟩ ∈ (ℝ^* × ℝ^*))
4		ltrelxr 11394	. . . . 5 ⊢ < ⊆ (ℝ^* × ℝ^*)
5	4	sseli 3909	. . . 4 ⊢ (⟨𝐴, 𝐴⟩ ∈ < → ⟨𝐴, 𝐴⟩ ∈ (ℝ^* × ℝ^*))
6	3, 5	nsyl 141	. . 3 ⊢ (¬ 𝐴 ∈ ℝ^* → ¬ ⟨𝐴, 𝐴⟩ ∈ < )
7		df-br 5107	. . 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 2147 ⟨cop 4838 class class class wbr 5106 × cxp 5653 ℝ^*cxr 11366 < clt 11367

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-pre-lttri 11298 ax-pre-lttrn 11299

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-er 8742 df-en 8999 df-dom 9000 df-sdom 9001 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372

This theorem is used by: (None)

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