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Mirrors > Home > MPE Home > Th. List > sltne | Structured version Visualization version GIF version |
Description: Surreal less-than implies not equal. (Contributed by Scott Fenton, 12-Mar-2025.) |
Ref | Expression |
---|---|
sltne | ⊢ ((𝐴 ∈ No ∧ 𝐴 <s 𝐵) → 𝐵 ≠ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sltirr 27672 | . . . 4 ⊢ (𝐴 ∈ No → ¬ 𝐴 <s 𝐴) | |
2 | breq2 5146 | . . . . 5 ⊢ (𝐵 = 𝐴 → (𝐴 <s 𝐵 ↔ 𝐴 <s 𝐴)) | |
3 | 2 | notbid 318 | . . . 4 ⊢ (𝐵 = 𝐴 → (¬ 𝐴 <s 𝐵 ↔ ¬ 𝐴 <s 𝐴)) |
4 | 1, 3 | syl5ibrcom 246 | . . 3 ⊢ (𝐴 ∈ No → (𝐵 = 𝐴 → ¬ 𝐴 <s 𝐵)) |
5 | 4 | necon2ad 2951 | . 2 ⊢ (𝐴 ∈ No → (𝐴 <s 𝐵 → 𝐵 ≠ 𝐴)) |
6 | 5 | imp 406 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐴 <s 𝐵) → 𝐵 ≠ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 class class class wbr 5142 No csur 27566 <s cslt 27567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-1o 8480 df-2o 8481 df-no 27569 df-slt 27570 |
This theorem is referenced by: sltlend 27697 sgt0ne0 27760 0elright 27830 |
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