![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 1nns | Structured version Visualization version GIF version |
Description: Surreal one is a positive surreal integer. (Contributed by Scott Fenton, 15-Apr-2025.) |
Ref | Expression |
---|---|
1nns | ⊢ 1s ∈ ℕs |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1n0s 28234 | . . 3 ⊢ 1s ∈ ℕ0s | |
2 | 0slt1s 27782 | . . . 4 ⊢ 0s <s 1s | |
3 | sgt0ne0 27787 | . . . 4 ⊢ ( 0s <s 1s → 1s ≠ 0s ) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ 1s ≠ 0s |
5 | eldifsn 4795 | . . 3 ⊢ ( 1s ∈ (ℕ0s ∖ { 0s }) ↔ ( 1s ∈ ℕ0s ∧ 1s ≠ 0s )) | |
6 | 1, 4, 5 | mpbir2an 709 | . 2 ⊢ 1s ∈ (ℕ0s ∖ { 0s }) |
7 | df-nns 28208 | . 2 ⊢ ℕs = (ℕ0s ∖ { 0s }) | |
8 | 6, 7 | eleqtrri 2828 | 1 ⊢ 1s ∈ ℕs |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ≠ wne 2937 ∖ cdif 3946 {csn 4632 class class class wbr 5152 <s cslt 27594 0s c0s 27775 1s c1s 27776 ℕ0scnn0s 28205 ℕscnns 28206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-no 27596 df-slt 27597 df-bday 27598 df-sle 27698 df-sslt 27734 df-scut 27736 df-0s 27777 df-1s 27778 df-made 27794 df-old 27795 df-left 27797 df-right 27798 df-norec2 27886 df-adds 27897 df-n0s 28207 df-nns 28208 |
This theorem is referenced by: peano2nns 28236 0reno 28245 remulscllem1 28248 |
Copyright terms: Public domain | W3C validator |