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Mathbox for Ender Ting |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > et-ltneverrefl | Structured version Visualization version GIF version |
Description: Less-than class is never reflexive. (Contributed by Ender Ting, 22-Nov-2024.) Prefer to specify theorem domain and then apply ltnri 11271. (New usage is discouraged.) |
Ref | Expression |
---|---|
et-ltneverrefl | ⊢ ¬ 𝐴 < 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltnr 13047 | . 2 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴) | |
2 | opelxp1 5679 | . . . . 5 ⊢ (⟨𝐴, 𝐴⟩ ∈ (ℝ* × ℝ*) → 𝐴 ∈ ℝ*) | |
3 | 2 | con3i 154 | . . . 4 ⊢ (¬ 𝐴 ∈ ℝ* → ¬ ⟨𝐴, 𝐴⟩ ∈ (ℝ* × ℝ*)) |
4 | ltrelxr 11223 | . . . . 5 ⊢ < ⊆ (ℝ* × ℝ*) | |
5 | 4 | sseli 3945 | . . . 4 ⊢ (⟨𝐴, 𝐴⟩ ∈ < → ⟨𝐴, 𝐴⟩ ∈ (ℝ* × ℝ*)) |
6 | 3, 5 | nsyl 140 | . . 3 ⊢ (¬ 𝐴 ∈ ℝ* → ¬ ⟨𝐴, 𝐴⟩ ∈ < ) |
7 | df-br 5111 | . . 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 2107 ⟨cop 4597 class class class wbr 5110 × cxp 5636 ℝ*cxr 11195 < clt 11196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-pre-lttri 11132 ax-pre-lttrn 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 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 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 |
This theorem is referenced by: tworepnotupword 45199 |
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