et-ltneverrefl - Mathbox for Ender Ting

	Mathbox for Ender Ting		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > et-ltneverrefl		Structured version Visualization version GIF version

Theorem et-ltneverrefl 47326

Description: Less-than class is never reflexive. (Contributed by Ender Ting, 22-Nov-2024.) Prefer to specify theorem domain and then apply ltnri 11251. (New usage is discouraged.)

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

**Proof of Theorem et-ltneverrefl**
Step	Hyp	Ref	Expression
1		xrltnr 13065	. 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 11202	. . . . 5 ⊢ < ⊆ (ℝ^* × ℝ^*)
5	4	sseli 3912	. . . 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 331	. 2 ⊢ (¬ 𝐴 ∈ ℝ^* → ¬ 𝐴 < 𝐴)
9	1, 8	pm2.61i 183	1 ⊢ ¬ 𝐴 < 𝐴

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∈ wcel 2121 ⟨cop 4563 class class class wbr 5074 × cxp 5618 ℝ^*cxr 11174 < clt 11175

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-cnex 11090 ax-resscn 11091 ax-pre-lttri 11108 ax-pre-lttrn 11109

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 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 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180

This theorem is referenced by: (None)

W3C validator