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Theorem 1p1e2s 28400
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.
StepHypRef Expression
1 0sno 27871 . . . . . . . . . 10 0s No
21elexi 3503 . . . . . . . . 9 0s ∈ V
3 oveq1 7438 . . . . . . . . . 10 (𝑦 = 0s → (𝑦 +s 1s ) = ( 0s +s 1s ))
43eqeq2d 2748 . . . . . . . . 9 (𝑦 = 0s → (𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = ( 0s +s 1s )))
52, 4rexsn 4682 . . . . . . . 8 (∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = ( 0s +s 1s ))
6 1sno 27872 . . . . . . . . . 10 1s No
7 addslid 28001 . . . . . . . . . 10 ( 1s No → ( 0s +s 1s ) = 1s )
86, 7ax-mp 5 . . . . . . . . 9 ( 0s +s 1s ) = 1s
98eqeq2i 2750 . . . . . . . 8 (𝑥 = ( 0s +s 1s ) ↔ 𝑥 = 1s )
105, 9bitri 275 . . . . . . 7 (∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s ) ↔ 𝑥 = 1s )
1110abbii 2809 . . . . . 6 {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} = {𝑥𝑥 = 1s }
12 df-sn 4627 . . . . . 6 { 1s } = {𝑥𝑥 = 1s }
1311, 12eqtr4i 2768 . . . . 5 {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} = { 1s }
14 oveq2 7439 . . . . . . . . . 10 (𝑦 = 0s → ( 1s +s 𝑦) = ( 1s +s 0s ))
1514eqeq2d 2748 . . . . . . . . 9 (𝑦 = 0s → (𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = ( 1s +s 0s )))
162, 15rexsn 4682 . . . . . . . 8 (∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = ( 1s +s 0s ))
17 addsrid 27997 . . . . . . . . . 10 ( 1s No → ( 1s +s 0s ) = 1s )
186, 17ax-mp 5 . . . . . . . . 9 ( 1s +s 0s ) = 1s
1918eqeq2i 2750 . . . . . . . 8 (𝑥 = ( 1s +s 0s ) ↔ 𝑥 = 1s )
2016, 19bitri 275 . . . . . . 7 (∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦) ↔ 𝑥 = 1s )
2120abbii 2809 . . . . . 6 {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)} = {𝑥𝑥 = 1s }
2221, 12eqtr4i 2768 . . . . 5 {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)} = { 1s }
2313, 22uneq12i 4166 . . . 4 ({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) = ({ 1s } ∪ { 1s })
24 unidm 4157 . . . 4 ({ 1s } ∪ { 1s }) = { 1s }
2523, 24eqtri 2765 . . 3 ({𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ { 0s }𝑥 = ( 1s +s 𝑦)}) = { 1s }
26 rex0 4360 . . . . . 6 ¬ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )
2726abf 4406 . . . . 5 {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} = ∅
28 rex0 4360 . . . . . 6 ¬ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)
2928abf 4406 . . . . 5 {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)} = ∅
3027, 29uneq12i 4166 . . . 4 ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)}) = (∅ ∪ ∅)
31 unidm 4157 . . . 4 (∅ ∪ ∅) = ∅
3230, 31eqtri 2765 . . 3 ({𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = (𝑦 +s 1s )} ∪ {𝑥 ∣ ∃𝑦 ∈ ∅ 𝑥 = ( 1s +s 𝑦)}) = ∅
3325, 32oveq12i 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 5448 . . . . . . 7 ( 0s No → { 0s } ∈ 𝒫 No )
351, 34ax-mp 5 . . . . . 6 { 0s } ∈ 𝒫 No
36 nulssgt 27843 . . . . . 6 ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
3735, 36ax-mp 5 . . . . 5 { 0s } <<s ∅
3837a1i 11 . . . 4 (⊤ → { 0s } <<s ∅)
39 df-1s 27870 . . . . 5 1s = ({ 0s } |s ∅)
4039a1i 11 . . . 4 (⊤ → 1s = ({ 0s } |s ∅))
4138, 38, 40, 40addsunif 28035 . . 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)})))
4241mptru 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 28395 . 2 2s = ({ 1s } |s ∅)
4433, 42, 433eqtr4i 2775 1 ( 1s +s 1s ) = 2s
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wtru 1541  wcel 2108  {cab 2714  wrex 3070  cun 3949  c0 4333  𝒫 cpw 4600  {csn 4626   class class class wbr 5143  (class class class)co 7431   No csur 27684   <<s csslt 27825   |s cscut 27827   0s c0s 27867   1s c1s 27868   +s cadds 27992  2sc2s 28394
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec2 27982  df-adds 27993  df-2s 28395
This theorem is referenced by:  no2times  28401  2nns  28402  n0seo  28405  zseo  28406  addhalfcut  28419  zs12bday  28424
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