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Theorem made0 34057
Description: The only surreal made on day is 0s. (Contributed by Scott Fenton, 7-Aug-2024.)
Assertion
Ref Expression
made0 ( M ‘∅) = { 0s }

Proof of Theorem made0
Dummy variables 𝑥 𝑙 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0elon 6319 . . 3 ∅ ∈ On
2 madeval2 34037 . . 3 (∅ ∈ On → ( M ‘∅) = {𝑥 No ∣ ∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)})
31, 2ax-mp 5 . 2 ( M ‘∅) = {𝑥 No ∣ ∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)}
4 rabeqsn 4602 . . 3 ({𝑥 No ∣ ∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)} = { 0s } ↔ ∀𝑥((𝑥 No ∧ ∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ 𝑥 = 0s ))
5 0elpw 5278 . . . . . . . 8 ∅ ∈ 𝒫 No
6 nulssgt 33992 . . . . . . . 8 (∅ ∈ 𝒫 No → ∅ <<s ∅)
75, 6ax-mp 5 . . . . . . 7 ∅ <<s ∅
8 ima0 5985 . . . . . . . . . . . . 13 ( M “ ∅) = ∅
98unieqi 4852 . . . . . . . . . . . 12 ( M “ ∅) =
10 uni0 4869 . . . . . . . . . . . 12 ∅ = ∅
119, 10eqtri 2766 . . . . . . . . . . 11 ( M “ ∅) = ∅
1211pweqi 4551 . . . . . . . . . 10 𝒫 ( M “ ∅) = 𝒫 ∅
13 pw0 4745 . . . . . . . . . 10 𝒫 ∅ = {∅}
1412, 13eqtri 2766 . . . . . . . . 9 𝒫 ( M “ ∅) = {∅}
1514rexeqi 3347 . . . . . . . 8 (∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅}∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
1614rexeqi 3347 . . . . . . . . 9 (∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
1716rexbii 3181 . . . . . . . 8 (∃𝑙 ∈ {∅}∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
18 0ex 5231 . . . . . . . . . . 11 ∅ ∈ V
19 breq2 5078 . . . . . . . . . . . 12 (𝑟 = ∅ → (𝑙 <<s 𝑟𝑙 <<s ∅))
20 oveq2 7283 . . . . . . . . . . . . 13 (𝑟 = ∅ → (𝑙 |s 𝑟) = (𝑙 |s ∅))
2120eqeq1d 2740 . . . . . . . . . . . 12 (𝑟 = ∅ → ((𝑙 |s 𝑟) = 𝑥 ↔ (𝑙 |s ∅) = 𝑥))
2219, 21anbi12d 631 . . . . . . . . . . 11 (𝑟 = ∅ → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥)))
2318, 22rexsn 4618 . . . . . . . . . 10 (∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
2423rexbii 3181 . . . . . . . . 9 (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
25 breq1 5077 . . . . . . . . . . 11 (𝑙 = ∅ → (𝑙 <<s ∅ ↔ ∅ <<s ∅))
26 oveq1 7282 . . . . . . . . . . . 12 (𝑙 = ∅ → (𝑙 |s ∅) = (∅ |s ∅))
2726eqeq1d 2740 . . . . . . . . . . 11 (𝑙 = ∅ → ((𝑙 |s ∅) = 𝑥 ↔ (∅ |s ∅) = 𝑥))
2825, 27anbi12d 631 . . . . . . . . . 10 (𝑙 = ∅ → ((𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥)))
2918, 28rexsn 4618 . . . . . . . . 9 (∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
3024, 29bitri 274 . . . . . . . 8 (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
3115, 17, 303bitri 297 . . . . . . 7 (∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
327, 31mpbiran 706 . . . . . 6 (∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ |s ∅) = 𝑥)
33 df-0s 34018 . . . . . . 7 0s = (∅ |s ∅)
3433eqeq1i 2743 . . . . . 6 ( 0s = 𝑥 ↔ (∅ |s ∅) = 𝑥)
35 eqcom 2745 . . . . . 6 ( 0s = 𝑥𝑥 = 0s )
3632, 34, 353bitr2i 299 . . . . 5 (∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ 𝑥 = 0s )
3736anbi2i 623 . . . 4 ((𝑥 No ∧ ∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ (𝑥 No 𝑥 = 0s ))
38 0sno 34020 . . . . . 6 0s ∈ No
39 eleq1 2826 . . . . . 6 (𝑥 = 0s → (𝑥 No ↔ 0s ∈ No ))
4038, 39mpbiri 257 . . . . 5 (𝑥 = 0s → 𝑥 No )
4140pm4.71ri 561 . . . 4 (𝑥 = 0s ↔ (𝑥 No 𝑥 = 0s ))
4237, 41bitr4i 277 . . 3 ((𝑥 No ∧ ∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ 𝑥 = 0s )
434, 42mpgbir 1802 . 2 {𝑥 No ∣ ∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)} = { 0s }
443, 43eqtri 2766 1 ( M ‘∅) = { 0s }
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1539  wcel 2106  wrex 3065  {crab 3068  c0 4256  𝒫 cpw 4533  {csn 4561   cuni 4839   class class class wbr 5074  cima 5592  Oncon0 6266  cfv 6433  (class class class)co 7275   No csur 33843   <<s csslt 33975   |s cscut 33977   0s c0s 34016   M cmade 34026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848  df-sslt 33976  df-scut 33978  df-0s 34018  df-made 34031
This theorem is referenced by:  new0  34058
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