Theorem 1p1e2s 28359

Description: One plus one is two. Surreal version. (Contributed by Scott Fenton, 27-May-2025.)

Assertion

Ref	Expression
1p1e2s	⊢ ( 1s +s 1s ) = 2s

Proof of Theorem 1p1e2s

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0sno 27795	. . . . . . . . . 10 ⊢ 0s ∈ No
2	1	elexi 3487	. . . . . . . . 9 ⊢ 0s ∈ V
3		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑦 = 0s → (𝑦 +s 1s ) = ( 0s +s 1s ))
4	3	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑦 = 0s → (𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = ( 0s +s 1s )))
5	2, 4	rexsn 4663	. . . . . . . 8 ⊢ (∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = ( 0s +s 1s ))
6		1sno 27796	. . . . . . . . . 10 ⊢ 1s ∈ No
7		addslid 27932	. . . . . . . . . 10 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
8	6, 7	ax-mp 5	. . . . . . . . 9 ⊢ ( 0s +s 1s ) = 1s
9	8	eqeq2i 2749	. . . . . . . 8 ⊢ (𝑥 = ( 0s +s 1s ) ↔ 𝑥 = 1s )
10	5, 9	bitri 275	. . . . . . 7 ⊢ (∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = 1s )
11	10	abbii 2803	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} = {𝑥 ∣ 𝑥 = 1s }
12		df-sn 4607	. . . . . 6 ⊢ { 1s } = {𝑥 ∣ 𝑥 = 1s }
13	11, 12	eqtr4i 2762	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} = { 1s }
14		oveq2 7418	. . . . . . . . . 10 ⊢ (𝑦 = 0s → ( 1s +s 𝑦) = ( 1s +s 0s ))
15	14	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑦 = 0s → (𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = ( 1s +s 0s )))
16	2, 15	rexsn 4663	. . . . . . . 8 ⊢ (∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = ( 1s +s 0s ))
17		addsrid 27928	. . . . . . . . . 10 ⊢ ( 1s ∈ No → ( 1s +s 0s ) = 1s )
18	6, 17	ax-mp 5	. . . . . . . . 9 ⊢ ( 1s +s 0s ) = 1s
19	18	eqeq2i 2749	. . . . . . . 8 ⊢ (𝑥 = ( 1s +s 0s ) ↔ 𝑥 = 1s )
20	16, 19	bitri 275	. . . . . . 7 ⊢ (∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = 1s )
21	20	abbii 2803	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)} = {𝑥 ∣ 𝑥 = 1s }
22	21, 12	eqtr4i 2762	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)} = { 1s }
23	13, 22	uneq12i 4146	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) = ({ 1s } ∪ { 1s })
24		unidm 4137	. . . 4 ⊢ ({ 1s } ∪ { 1s }) = { 1s }
25	23, 24	eqtri 2759	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) = { 1s }
26		rex0 4340	. . . . . 6 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )
27	26	abf 4386	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} = ∅
28		rex0 4340	. . . . . 6 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)
29	28	abf 4386	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)} = ∅
30	27, 29	uneq12i 4146	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)}) = (∅ ∪ ∅)
31		unidm 4137	. . . 4 ⊢ (∅ ∪ ∅) = ∅
32	30, 31	eqtri 2759	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)}) = ∅
33	25, 32	oveq12i 7422	. 2 ⊢ (({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) |s ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)})) = ({ 1s } |s ∅)
34		snelpwi 5423	. . . . . . 7 ⊢ ( 0s ∈ No → { 0s } ∈ 𝒫 No )
35	1, 34	ax-mp 5	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
36		nulssgt 27767	. . . . . 6 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
37	35, 36	ax-mp 5	. . . . 5 ⊢ { 0s } <<s ∅
38	37	a1i 11	. . . 4 ⊢ (⊤ → { 0s } <<s ∅)
39		df-1s 27794	. . . . 5 ⊢ 1s = ({ 0s } |s ∅)
40	39	a1i 11	. . . 4 ⊢ (⊤ → 1s = ({ 0s } |s ∅))
41	38, 38, 40, 40	addsunif 27966	. . 3 ⊢ (⊤ → ( 1s +s 1s ) = (({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) |s ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)})))
42	41	mptru 1547	. 2 ⊢ ( 1s +s 1s ) = (({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) |s ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)}))
43		df-2s 28354	. 2 ⊢ 2s = ({ 1s } |s ∅)
44	33, 42, 43	3eqtr4i 2769	1 ⊢ ( 1s +s 1s ) = 2s

Syntax hints:   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  {cab 2714  ∃wrex 3061   ∪ cun 3929  ∅c0 4313  𝒫 cpw 4580  {csn 4606   class class class wbr 5124  (class class class)co 7410   No csur 27608   <<s csslt 27749   |s cscut 27751   0_s c0s 27791   1_s c1s 27792   +_s cadds 27923  2_sc2s 28353

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-ot 4615  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-1o 8485  df-2o 8486  df-nadd 8683  df-no 27611  df-slt 27612  df-bday 27613  df-sle 27714  df-sslt 27750  df-scut 27752  df-0s 27793  df-1s 27794  df-made 27812  df-old 27813  df-left 27815  df-right 27816  df-norec2 27913  df-adds 27924  df-2s 28354

This theorem is referenced by:  no2times  28360  2nns  28361  n0seo  28364  zseo  28365  addhalfcut  28391  pw2cutp1  28393  zs12bday  28400