Theorem 1p1e2s 28418

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 27889	. . . . . . . . . 10 ⊢ 0s ∈ No
2	1	elexi 3511	. . . . . . . . 9 ⊢ 0s ∈ V
3		oveq1 7455	. . . . . . . . . 10 ⊢ (𝑦 = 0s → (𝑦 +s 1s ) = ( 0s +s 1s ))
4	3	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑦 = 0s → (𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = ( 0s +s 1s )))
5	2, 4	rexsn 4706	. . . . . . . 8 ⊢ (∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = ( 0s +s 1s ))
6		1sno 27890	. . . . . . . . . 10 ⊢ 1s ∈ No
7		addslid 28019	. . . . . . . . . 10 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
8	6, 7	ax-mp 5	. . . . . . . . 9 ⊢ ( 0s +s 1s ) = 1s
9	8	eqeq2i 2753	. . . . . . . 8 ⊢ (𝑥 = ( 0s +s 1s ) ↔ 𝑥 = 1s )
10	5, 9	bitri 275	. . . . . . 7 ⊢ (∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = 1s )
11	10	abbii 2812	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} = {𝑥 ∣ 𝑥 = 1s }
12		df-sn 4649	. . . . . 6 ⊢ { 1s } = {𝑥 ∣ 𝑥 = 1s }
13	11, 12	eqtr4i 2771	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} = { 1s }
14		oveq2 7456	. . . . . . . . . 10 ⊢ (𝑦 = 0s → ( 1s +s 𝑦) = ( 1s +s 0s ))
15	14	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑦 = 0s → (𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = ( 1s +s 0s )))
16	2, 15	rexsn 4706	. . . . . . . 8 ⊢ (∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = ( 1s +s 0s ))
17		addsrid 28015	. . . . . . . . . 10 ⊢ ( 1s ∈ No → ( 1s +s 0s ) = 1s )
18	6, 17	ax-mp 5	. . . . . . . . 9 ⊢ ( 1s +s 0s ) = 1s
19	18	eqeq2i 2753	. . . . . . . 8 ⊢ (𝑥 = ( 1s +s 0s ) ↔ 𝑥 = 1s )
20	16, 19	bitri 275	. . . . . . 7 ⊢ (∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = 1s )
21	20	abbii 2812	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)} = {𝑥 ∣ 𝑥 = 1s }
22	21, 12	eqtr4i 2771	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)} = { 1s }
23	13, 22	uneq12i 4189	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) = ({ 1s } ∪ { 1s })
24		unidm 4180	. . . 4 ⊢ ({ 1s } ∪ { 1s }) = { 1s }
25	23, 24	eqtri 2768	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) = { 1s }
26		rex0 4383	. . . . . 6 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )
27	26	abf 4429	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} = ∅
28		rex0 4383	. . . . . 6 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)
29	28	abf 4429	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)} = ∅
30	27, 29	uneq12i 4189	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)}) = (∅ ∪ ∅)
31		unidm 4180	. . . 4 ⊢ (∅ ∪ ∅) = ∅
32	30, 31	eqtri 2768	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)}) = ∅
33	25, 32	oveq12i 7460	. 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 5463	. . . . . . 7 ⊢ ( 0s ∈ No → { 0s } ∈ 𝒫 No )
35	1, 34	ax-mp 5	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
36		nulssgt 27861	. . . . . 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 27888	. . . . 5 ⊢ 1s = ({ 0s } |s ∅)
40	39	a1i 11	. . . 4 ⊢ (⊤ → 1s = ({ 0s } |s ∅))
41	38, 38, 40, 40	addsunif 28053	. . 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 28413	. 2 ⊢ 2s = ({ 1s } |s ∅)
44	33, 42, 43	3eqtr4i 2778	1 ⊢ ( 1s +s 1s ) = 2s

Syntax hints:   = wceq 1537  ⊤wtru 1538   ∈ wcel 2108  {cab 2717  ∃wrex 3076   ∪ cun 3974  ∅c0 4352  𝒫 cpw 4622  {csn 4648   class class class wbr 5166  (class class class)co 7448   No csur 27702   <<s csslt 27843   |s cscut 27845   0_s c0s 27885   1_s c1s 27886   +_s cadds 28010  2_sc2s 28412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec2 28000  df-adds 28011  df-2s 28413

This theorem is referenced by:  no2times  28419  2nns  28420  n0seo  28423  zseo  28424  addhalfcut  28437  zs12bday  28442