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Theorem made0 27912
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 6438 . . 3 ∅ ∈ On
2 madeval2 27892 . . 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 4667 . . 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 5356 . . . . . . . 8 ∅ ∈ 𝒫 No
6 nulssgt 27843 . . . . . . . 8 (∅ ∈ 𝒫 No → ∅ <<s ∅)
75, 6ax-mp 5 . . . . . . 7 ∅ <<s ∅
8 ima0 6095 . . . . . . . . . . . . 13 ( M “ ∅) = ∅
98unieqi 4919 . . . . . . . . . . . 12 ( M “ ∅) =
10 uni0 4935 . . . . . . . . . . . 12 ∅ = ∅
119, 10eqtri 2765 . . . . . . . . . . 11 ( M “ ∅) = ∅
1211pweqi 4616 . . . . . . . . . 10 𝒫 ( M “ ∅) = 𝒫 ∅
13 pw0 4812 . . . . . . . . . 10 𝒫 ∅ = {∅}
1412, 13eqtri 2765 . . . . . . . . 9 𝒫 ( M “ ∅) = {∅}
1514rexeqi 3325 . . . . . . . 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 3325 . . . . . . . . 9 (∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
1716rexbii 3094 . . . . . . . 8 (∃𝑙 ∈ {∅}∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
18 0ex 5307 . . . . . . . . . . 11 ∅ ∈ V
19 breq2 5147 . . . . . . . . . . . 12 (𝑟 = ∅ → (𝑙 <<s 𝑟𝑙 <<s ∅))
20 oveq2 7439 . . . . . . . . . . . . 13 (𝑟 = ∅ → (𝑙 |s 𝑟) = (𝑙 |s ∅))
2120eqeq1d 2739 . . . . . . . . . . . 12 (𝑟 = ∅ → ((𝑙 |s 𝑟) = 𝑥 ↔ (𝑙 |s ∅) = 𝑥))
2219, 21anbi12d 632 . . . . . . . . . . 11 (𝑟 = ∅ → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥)))
2318, 22rexsn 4682 . . . . . . . . . 10 (∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
2423rexbii 3094 . . . . . . . . 9 (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
25 breq1 5146 . . . . . . . . . . 11 (𝑙 = ∅ → (𝑙 <<s ∅ ↔ ∅ <<s ∅))
26 oveq1 7438 . . . . . . . . . . . 12 (𝑙 = ∅ → (𝑙 |s ∅) = (∅ |s ∅))
2726eqeq1d 2739 . . . . . . . . . . 11 (𝑙 = ∅ → ((𝑙 |s ∅) = 𝑥 ↔ (∅ |s ∅) = 𝑥))
2825, 27anbi12d 632 . . . . . . . . . 10 (𝑙 = ∅ → ((𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥)))
2918, 28rexsn 4682 . . . . . . . . 9 (∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
3024, 29bitri 275 . . . . . . . 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 709 . . . . . 6 (∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ |s ∅) = 𝑥)
33 df-0s 27869 . . . . . . 7 0s = (∅ |s ∅)
3433eqeq1i 2742 . . . . . 6 ( 0s = 𝑥 ↔ (∅ |s ∅) = 𝑥)
35 eqcom 2744 . . . . . 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 27871 . . . . . 6 0s No
39 eleq1 2829 . . . . . 6 (𝑥 = 0s → (𝑥 No ↔ 0s No ))
4038, 39mpbiri 258 . . . . 5 (𝑥 = 0s𝑥 No )
4140pm4.71ri 560 . . . 4 (𝑥 = 0s ↔ (𝑥 No 𝑥 = 0s ))
4237, 41bitr4i 278 . . 3 ((𝑥 No ∧ ∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ 𝑥 = 0s )
434, 42mpgbir 1799 . 2 {𝑥 No ∣ ∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)} = { 0s }
443, 43eqtri 2765 1 ( M ‘∅) = { 0s }
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  {crab 3436  c0 4333  𝒫 cpw 4600  {csn 4626   cuni 4907   class class class wbr 5143  cima 5688  Oncon0 6384  cfv 6561  (class class class)co 7431   No csur 27684   <<s csslt 27825   |s cscut 27827   0s c0s 27867   M cmade 27881
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-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-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-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886
This theorem is referenced by:  new0  27913  old1  27914
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