Theorem 1p1e2s 28309

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 27745	. . . . . . . . . 10 ⊢ 0s ∈ No
2	1	elexi 3473	. . . . . . . . 9 ⊢ 0s ∈ V
3		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑦 = 0s → (𝑦 +s 1s ) = ( 0s +s 1s ))
4	3	eqeq2d 2741	. . . . . . . . 9 ⊢ (𝑦 = 0s → (𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = ( 0s +s 1s )))
5	2, 4	rexsn 4649	. . . . . . . 8 ⊢ (∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = ( 0s +s 1s ))
6		1sno 27746	. . . . . . . . . 10 ⊢ 1s ∈ No
7		addslid 27882	. . . . . . . . . 10 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
8	6, 7	ax-mp 5	. . . . . . . . 9 ⊢ ( 0s +s 1s ) = 1s
9	8	eqeq2i 2743	. . . . . . . 8 ⊢ (𝑥 = ( 0s +s 1s ) ↔ 𝑥 = 1s )
10	5, 9	bitri 275	. . . . . . 7 ⊢ (∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = 1s )
11	10	abbii 2797	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} = {𝑥 ∣ 𝑥 = 1s }
12		df-sn 4593	. . . . . 6 ⊢ { 1s } = {𝑥 ∣ 𝑥 = 1s }
13	11, 12	eqtr4i 2756	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} = { 1s }
14		oveq2 7398	. . . . . . . . . 10 ⊢ (𝑦 = 0s → ( 1s +s 𝑦) = ( 1s +s 0s ))
15	14	eqeq2d 2741	. . . . . . . . 9 ⊢ (𝑦 = 0s → (𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = ( 1s +s 0s )))
16	2, 15	rexsn 4649	. . . . . . . 8 ⊢ (∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = ( 1s +s 0s ))
17		addsrid 27878	. . . . . . . . . 10 ⊢ ( 1s ∈ No → ( 1s +s 0s ) = 1s )
18	6, 17	ax-mp 5	. . . . . . . . 9 ⊢ ( 1s +s 0s ) = 1s
19	18	eqeq2i 2743	. . . . . . . 8 ⊢ (𝑥 = ( 1s +s 0s ) ↔ 𝑥 = 1s )
20	16, 19	bitri 275	. . . . . . 7 ⊢ (∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = 1s )
21	20	abbii 2797	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)} = {𝑥 ∣ 𝑥 = 1s }
22	21, 12	eqtr4i 2756	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)} = { 1s }
23	13, 22	uneq12i 4132	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) = ({ 1s } ∪ { 1s })
24		unidm 4123	. . . 4 ⊢ ({ 1s } ∪ { 1s }) = { 1s }
25	23, 24	eqtri 2753	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) = { 1s }
26		rex0 4326	. . . . . 6 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )
27	26	abf 4372	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} = ∅
28		rex0 4326	. . . . . 6 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)
29	28	abf 4372	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)} = ∅
30	27, 29	uneq12i 4132	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)}) = (∅ ∪ ∅)
31		unidm 4123	. . . 4 ⊢ (∅ ∪ ∅) = ∅
32	30, 31	eqtri 2753	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)}) = ∅
33	25, 32	oveq12i 7402	. 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 5406	. . . . . . 7 ⊢ ( 0s ∈ No → { 0s } ∈ 𝒫 No )
35	1, 34	ax-mp 5	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
36		nulssgt 27717	. . . . . 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 27744	. . . . 5 ⊢ 1s = ({ 0s } |s ∅)
40	39	a1i 11	. . . 4 ⊢ (⊤ → 1s = ({ 0s } |s ∅))
41	38, 38, 40, 40	addsunif 27916	. . 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 28304	. 2 ⊢ 2s = ({ 1s } |s ∅)
44	33, 42, 43	3eqtr4i 2763	1 ⊢ ( 1s +s 1s ) = 2s

Syntax hints:   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  {cab 2708  ∃wrex 3054   ∪ cun 3915  ∅c0 4299  𝒫 cpw 4566  {csn 4592   class class class wbr 5110  (class class class)co 7390   No csur 27558   <<s csslt 27699   |s cscut 27701   0_s c0s 27741   1_s c1s 27742   +_s cadds 27873  2_sc2s 28303

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-1o 8437  df-2o 8438  df-nadd 8633  df-no 27561  df-slt 27562  df-bday 27563  df-sle 27664  df-sslt 27700  df-scut 27702  df-0s 27743  df-1s 27744  df-made 27762  df-old 27763  df-left 27765  df-right 27766  df-norec2 27863  df-adds 27874  df-2s 28304

This theorem is referenced by:  no2times  28310  2nns  28311  n0seo  28314  zseo  28315  addhalfcut  28341  pw2cutp1  28343  zs12bday  28350