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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ltex | Structured version Visualization version GIF version | ||
| Description: The less-than relation is a set. (Contributed by SN, 5-Jun-2025.) |
| Ref | Expression |
|---|---|
| ltex | ⊢ < ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xrex 13029 | . . 3 ⊢ ℝ* ∈ V | |
| 2 | 1, 1 | xpex 7773 | . 2 ⊢ (ℝ* × ℝ*) ∈ V |
| 3 | ltrelxr 11322 | . 2 ⊢ < ⊆ (ℝ* × ℝ*) | |
| 4 | 2, 3 | ssexi 5322 | 1 ⊢ < ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Vcvv 3480 × cxp 5683 ℝ*cxr 11294 < clt 11295 |
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-opab 5206 df-xp 5691 df-rel 5692 df-xr 11299 df-ltxr 11300 |
| This theorem is referenced by: (None) |
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