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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrltned | Structured version Visualization version GIF version |
Description: 'Less than' implies not equal. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
Ref | Expression |
---|---|
xrltned.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xrltned.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
xrltned.3 | ⊢ (𝜑 → 𝐴 < 𝐵) |
Ref | Expression |
---|---|
xrltned | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltned.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | xrltned.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | xrltned.3 | . . 3 ⊢ (𝜑 → 𝐴 < 𝐵) | |
4 | 1, 2, 3 | xrgtned 41665 | . 2 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
5 | 4 | necomd 3070 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ≠ wne 3015 class class class wbr 5059 ℝ*cxr 10667 < clt 10668 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-pre-lttri 10604 ax-pre-lttrn 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 |
This theorem is referenced by: infxr 41710 infleinflem2 41714 xrpnf 41837 ge0lere 41883 liminflbuz2 42171 liminflimsupxrre 42173 ioorrnopnxrlem 42666 sge0hsphoire 42946 hoidmv1lelem1 42948 hoidmv1lelem2 42949 hoidmv1lelem3 42950 hoidmvlelem1 42952 hoidmvlelem4 42955 pimiooltgt 43064 |
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