Theorem 1p1e2s 28302

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 27738	. . . . . . . . . 10 ⊢ 0s ∈ No
2	1	elexi 3470	. . . . . . . . 9 ⊢ 0s ∈ V
3		oveq1 7394	. . . . . . . . . 10 ⊢ (𝑦 = 0s → (𝑦 +s 1s ) = ( 0s +s 1s ))
4	3	eqeq2d 2740	. . . . . . . . 9 ⊢ (𝑦 = 0s → (𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = ( 0s +s 1s )))
5	2, 4	rexsn 4646	. . . . . . . 8 ⊢ (∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = ( 0s +s 1s ))
6		1sno 27739	. . . . . . . . . 10 ⊢ 1s ∈ No
7		addslid 27875	. . . . . . . . . 10 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
8	6, 7	ax-mp 5	. . . . . . . . 9 ⊢ ( 0s +s 1s ) = 1s
9	8	eqeq2i 2742	. . . . . . . 8 ⊢ (𝑥 = ( 0s +s 1s ) ↔ 𝑥 = 1s )
10	5, 9	bitri 275	. . . . . . 7 ⊢ (∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = 1s )
11	10	abbii 2796	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} = {𝑥 ∣ 𝑥 = 1s }
12		df-sn 4590	. . . . . 6 ⊢ { 1s } = {𝑥 ∣ 𝑥 = 1s }
13	11, 12	eqtr4i 2755	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} = { 1s }
14		oveq2 7395	. . . . . . . . . 10 ⊢ (𝑦 = 0s → ( 1s +s 𝑦) = ( 1s +s 0s ))
15	14	eqeq2d 2740	. . . . . . . . 9 ⊢ (𝑦 = 0s → (𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = ( 1s +s 0s )))
16	2, 15	rexsn 4646	. . . . . . . 8 ⊢ (∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = ( 1s +s 0s ))
17		addsrid 27871	. . . . . . . . . 10 ⊢ ( 1s ∈ No → ( 1s +s 0s ) = 1s )
18	6, 17	ax-mp 5	. . . . . . . . 9 ⊢ ( 1s +s 0s ) = 1s
19	18	eqeq2i 2742	. . . . . . . 8 ⊢ (𝑥 = ( 1s +s 0s ) ↔ 𝑥 = 1s )
20	16, 19	bitri 275	. . . . . . 7 ⊢ (∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = 1s )
21	20	abbii 2796	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)} = {𝑥 ∣ 𝑥 = 1s }
22	21, 12	eqtr4i 2755	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)} = { 1s }
23	13, 22	uneq12i 4129	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) = ({ 1s } ∪ { 1s })
24		unidm 4120	. . . 4 ⊢ ({ 1s } ∪ { 1s }) = { 1s }
25	23, 24	eqtri 2752	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) = { 1s }
26		rex0 4323	. . . . . 6 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )
27	26	abf 4369	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} = ∅
28		rex0 4323	. . . . . 6 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)
29	28	abf 4369	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)} = ∅
30	27, 29	uneq12i 4129	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)}) = (∅ ∪ ∅)
31		unidm 4120	. . . 4 ⊢ (∅ ∪ ∅) = ∅
32	30, 31	eqtri 2752	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)}) = ∅
33	25, 32	oveq12i 7399	. 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 5403	. . . . . . 7 ⊢ ( 0s ∈ No → { 0s } ∈ 𝒫 No )
35	1, 34	ax-mp 5	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
36		nulssgt 27710	. . . . . 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 27737	. . . . 5 ⊢ 1s = ({ 0s } |s ∅)
40	39	a1i 11	. . . 4 ⊢ (⊤ → 1s = ({ 0s } |s ∅))
41	38, 38, 40, 40	addsunif 27909	. . 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 28297	. 2 ⊢ 2s = ({ 1s } |s ∅)
44	33, 42, 43	3eqtr4i 2762	1 ⊢ ( 1s +s 1s ) = 2s

Syntax hints:   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  {cab 2707  ∃wrex 3053   ∪ cun 3912  ∅c0 4296  𝒫 cpw 4563  {csn 4589   class class class wbr 5107  (class class class)co 7387   No csur 27551   <<s csslt 27692   |s cscut 27694   0_s c0s 27734   1_s c1s 27735   +_s cadds 27866  2_sc2s 28296

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-1o 8434  df-2o 8435  df-nadd 8630  df-no 27554  df-slt 27555  df-bday 27556  df-sle 27657  df-sslt 27693  df-scut 27695  df-0s 27736  df-1s 27737  df-made 27755  df-old 27756  df-left 27758  df-right 27759  df-norec2 27856  df-adds 27867  df-2s 28297

This theorem is referenced by:  no2times  28303  2nns  28304  n0seo  28307  zseo  28308  addhalfcut  28334  pw2cutp1  28336  zs12bday  28343