Theorem 1p1e2s 28415

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 27886	. . . . . . . . . 10 ⊢ 0s ∈ No
2	1	elexi 3501	. . . . . . . . 9 ⊢ 0s ∈ V
3		oveq1 7438	. . . . . . . . . 10 ⊢ (𝑦 = 0s → (𝑦 +s 1s ) = ( 0s +s 1s ))
4	3	eqeq2d 2746	. . . . . . . . 9 ⊢ (𝑦 = 0s → (𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = ( 0s +s 1s )))
5	2, 4	rexsn 4687	. . . . . . . 8 ⊢ (∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = ( 0s +s 1s ))
6		1sno 27887	. . . . . . . . . 10 ⊢ 1s ∈ No
7		addslid 28016	. . . . . . . . . 10 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
8	6, 7	ax-mp 5	. . . . . . . . 9 ⊢ ( 0s +s 1s ) = 1s
9	8	eqeq2i 2748	. . . . . . . 8 ⊢ (𝑥 = ( 0s +s 1s ) ↔ 𝑥 = 1s )
10	5, 9	bitri 275	. . . . . . 7 ⊢ (∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = 1s )
11	10	abbii 2807	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} = {𝑥 ∣ 𝑥 = 1s }
12		df-sn 4632	. . . . . 6 ⊢ { 1s } = {𝑥 ∣ 𝑥 = 1s }
13	11, 12	eqtr4i 2766	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} = { 1s }
14		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑦 = 0s → ( 1s +s 𝑦) = ( 1s +s 0s ))
15	14	eqeq2d 2746	. . . . . . . . 9 ⊢ (𝑦 = 0s → (𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = ( 1s +s 0s )))
16	2, 15	rexsn 4687	. . . . . . . 8 ⊢ (∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = ( 1s +s 0s ))
17		addsrid 28012	. . . . . . . . . 10 ⊢ ( 1s ∈ No → ( 1s +s 0s ) = 1s )
18	6, 17	ax-mp 5	. . . . . . . . 9 ⊢ ( 1s +s 0s ) = 1s
19	18	eqeq2i 2748	. . . . . . . 8 ⊢ (𝑥 = ( 1s +s 0s ) ↔ 𝑥 = 1s )
20	16, 19	bitri 275	. . . . . . 7 ⊢ (∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = 1s )
21	20	abbii 2807	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)} = {𝑥 ∣ 𝑥 = 1s }
22	21, 12	eqtr4i 2766	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)} = { 1s }
23	13, 22	uneq12i 4176	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) = ({ 1s } ∪ { 1s })
24		unidm 4167	. . . 4 ⊢ ({ 1s } ∪ { 1s }) = { 1s }
25	23, 24	eqtri 2763	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) = { 1s }
26		rex0 4366	. . . . . 6 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )
27	26	abf 4412	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} = ∅
28		rex0 4366	. . . . . 6 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)
29	28	abf 4412	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)} = ∅
30	27, 29	uneq12i 4176	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)}) = (∅ ∪ ∅)
31		unidm 4167	. . . 4 ⊢ (∅ ∪ ∅) = ∅
32	30, 31	eqtri 2763	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)}) = ∅
33	25, 32	oveq12i 7443	. 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 5454	. . . . . . 7 ⊢ ( 0s ∈ No → { 0s } ∈ 𝒫 No )
35	1, 34	ax-mp 5	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
36		nulssgt 27858	. . . . . 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 27885	. . . . 5 ⊢ 1s = ({ 0s } |s ∅)
40	39	a1i 11	. . . 4 ⊢ (⊤ → 1s = ({ 0s } |s ∅))
41	38, 38, 40, 40	addsunif 28050	. . 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 1544	. 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 28410	. 2 ⊢ 2s = ({ 1s } |s ∅)
44	33, 42, 43	3eqtr4i 2773	1 ⊢ ( 1s +s 1s ) = 2s

Syntax hints:   = wceq 1537  ⊤wtru 1538   ∈ wcel 2106  {cab 2712  ∃wrex 3068   ∪ cun 3961  ∅c0 4339  𝒫 cpw 4605  {csn 4631   class class class wbr 5148  (class class class)co 7431   No csur 27699   <<s csslt 27840   |s cscut 27842   0_s c0s 27882   1_s c1s 27883   +_s cadds 28007  2_sc2s 28409

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-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-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  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-2s 28410

This theorem is referenced by:  no2times  28416  2nns  28417  n0seo  28420  zseo  28421  addhalfcut  28434  zs12bday  28439