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Mirrors > Home > MPE Home > Th. List > sltlend | Structured version Visualization version GIF version |
Description: Surreal less-than in terms of less-than or equal. (Contributed by Scott Fenton, 15-Apr-2025.) |
Ref | Expression |
---|---|
sltlen.1 | ⊢ (𝜑 → 𝐴 ∈ No ) |
sltlen.2 | ⊢ (𝜑 → 𝐵 ∈ No ) |
Ref | Expression |
---|---|
sltlend | ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐴 ≤s 𝐵 ∧ 𝐵 ≠ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sltlen.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No ) | |
2 | 1 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 <s 𝐵) → 𝐴 ∈ No ) |
3 | sltlen.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No ) | |
4 | 3 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 <s 𝐵) → 𝐵 ∈ No ) |
5 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 <s 𝐵) → 𝐴 <s 𝐵) | |
6 | 2, 4, 5 | sltled 27689 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 <s 𝐵) → 𝐴 ≤s 𝐵) |
7 | 6 | ex 412 | . . 3 ⊢ (𝜑 → (𝐴 <s 𝐵 → 𝐴 ≤s 𝐵)) |
8 | sltne 27690 | . . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐴 <s 𝐵) → 𝐵 ≠ 𝐴) | |
9 | 1, 8 | sylan 579 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 <s 𝐵) → 𝐵 ≠ 𝐴) |
10 | 9 | ex 412 | . . 3 ⊢ (𝜑 → (𝐴 <s 𝐵 → 𝐵 ≠ 𝐴)) |
11 | 7, 10 | jcad 512 | . 2 ⊢ (𝜑 → (𝐴 <s 𝐵 → (𝐴 ≤s 𝐵 ∧ 𝐵 ≠ 𝐴))) |
12 | sleloe 27674 | . . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵))) | |
13 | 1, 3, 12 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵))) |
14 | eqneqall 2946 | . . . . . 6 ⊢ (𝐵 = 𝐴 → (𝐵 ≠ 𝐴 → 𝐴 <s 𝐵)) | |
15 | 14 | eqcoms 2735 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐵 ≠ 𝐴 → 𝐴 <s 𝐵)) |
16 | 15 | jao1i 857 | . . . 4 ⊢ ((𝐴 <s 𝐵 ∨ 𝐴 = 𝐵) → (𝐵 ≠ 𝐴 → 𝐴 <s 𝐵)) |
17 | 13, 16 | biimtrdi 252 | . . 3 ⊢ (𝜑 → (𝐴 ≤s 𝐵 → (𝐵 ≠ 𝐴 → 𝐴 <s 𝐵))) |
18 | 17 | impd 410 | . 2 ⊢ (𝜑 → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≠ 𝐴) → 𝐴 <s 𝐵)) |
19 | 11, 18 | impbid 211 | 1 ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐴 ≤s 𝐵 ∧ 𝐵 ≠ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 No csur 27560 <s cslt 27561 ≤s csle 27664 |
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 2164 ax-ext 2698 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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 27563 df-slt 27564 df-sle 27665 |
This theorem is referenced by: nnsgt0 28194 |
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