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Mirrors > Home > MPE Home > Th. List > nnsge1 | Structured version Visualization version GIF version |
Description: A positive surreal integer is greater than or equal to one. (Contributed by Scott Fenton, 26-Jul-2025.) |
Ref | Expression |
---|---|
nnsge1 | ⊢ (𝑁 ∈ ℕs → 1s ≤s 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnns 28358 | . 2 ⊢ (𝑁 ∈ ℕs ↔ (𝑁 ∈ ℕ0s ∧ 𝑁 ≠ 0s )) | |
2 | n0s0suc 28360 | . . 3 ⊢ (𝑁 ∈ ℕ0s → (𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s ))) | |
3 | neneq 2944 | . . 3 ⊢ (𝑁 ≠ 0s → ¬ 𝑁 = 0s ) | |
4 | pm2.53 851 | . . . . 5 ⊢ ((𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )) → (¬ 𝑁 = 0s → ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s ))) | |
5 | 4 | imp 406 | . . . 4 ⊢ (((𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )) ∧ ¬ 𝑁 = 0s ) → ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )) |
6 | 1sno 27887 | . . . . . . . 8 ⊢ 1s ∈ No | |
7 | addslid 28016 | . . . . . . . 8 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s ) | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 ⊢ ( 0s +s 1s ) = 1s |
9 | n0sge0 28356 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → 0s ≤s 𝑥) | |
10 | n0sno 28343 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ No ) | |
11 | 0sno 27886 | . . . . . . . . . 10 ⊢ 0s ∈ No | |
12 | sleadd1 28037 | . . . . . . . . . 10 ⊢ (( 0s ∈ No ∧ 𝑥 ∈ No ∧ 1s ∈ No ) → ( 0s ≤s 𝑥 ↔ ( 0s +s 1s ) ≤s (𝑥 +s 1s ))) | |
13 | 11, 6, 12 | mp3an13 1451 | . . . . . . . . 9 ⊢ (𝑥 ∈ No → ( 0s ≤s 𝑥 ↔ ( 0s +s 1s ) ≤s (𝑥 +s 1s ))) |
14 | 10, 13 | syl 17 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → ( 0s ≤s 𝑥 ↔ ( 0s +s 1s ) ≤s (𝑥 +s 1s ))) |
15 | 9, 14 | mpbid 232 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ0s → ( 0s +s 1s ) ≤s (𝑥 +s 1s )) |
16 | 8, 15 | eqbrtrrid 5184 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0s → 1s ≤s (𝑥 +s 1s )) |
17 | breq2 5152 | . . . . . 6 ⊢ (𝑁 = (𝑥 +s 1s ) → ( 1s ≤s 𝑁 ↔ 1s ≤s (𝑥 +s 1s ))) | |
18 | 16, 17 | syl5ibrcom 247 | . . . . 5 ⊢ (𝑥 ∈ ℕ0s → (𝑁 = (𝑥 +s 1s ) → 1s ≤s 𝑁)) |
19 | 18 | rexlimiv 3146 | . . . 4 ⊢ (∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s ) → 1s ≤s 𝑁) |
20 | 5, 19 | syl 17 | . . 3 ⊢ (((𝑁 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝑁 = (𝑥 +s 1s )) ∧ ¬ 𝑁 = 0s ) → 1s ≤s 𝑁) |
21 | 2, 3, 20 | syl2an 596 | . 2 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝑁 ≠ 0s ) → 1s ≤s 𝑁) |
22 | 1, 21 | sylbi 217 | 1 ⊢ (𝑁 ∈ ℕs → 1s ≤s 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 class class class wbr 5148 (class class class)co 7431 No csur 27699 ≤s csle 27804 0s c0s 27882 1s c1s 27883 +s cadds 28007 ℕ0scnn0s 28333 ℕscnns 28334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-nadd 8703 df-no 27702 df-slt 27703 df-bday 27704 df-sle 27805 df-sslt 27841 df-scut 27843 df-0s 27884 df-1s 27885 df-made 27901 df-old 27902 df-left 27904 df-right 27905 df-norec2 27997 df-adds 28008 df-n0s 28335 df-nns 28336 |
This theorem is referenced by: (None) |
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