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Theorem made0 33703
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 6226 . . 3 ∅ ∈ On
2 madeval2 33683 . . 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 4558 . . 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 5223 . . . . . . . 8 ∅ ∈ 𝒫 No
6 nulssgt 33638 . . . . . . . 8 (∅ ∈ 𝒫 No → ∅ <<s ∅)
75, 6ax-mp 5 . . . . . . 7 ∅ <<s ∅
8 ima0 5920 . . . . . . . . . . . . 13 ( M “ ∅) = ∅
98unieqi 4810 . . . . . . . . . . . 12 ( M “ ∅) =
10 uni0 4827 . . . . . . . . . . . 12 ∅ = ∅
119, 10eqtri 2762 . . . . . . . . . . 11 ( M “ ∅) = ∅
1211pweqi 4507 . . . . . . . . . 10 𝒫 ( M “ ∅) = 𝒫 ∅
13 pw0 4701 . . . . . . . . . 10 𝒫 ∅ = {∅}
1412, 13eqtri 2762 . . . . . . . . 9 𝒫 ( M “ ∅) = {∅}
1514rexeqi 3316 . . . . . . . 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 3316 . . . . . . . . 9 (∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
1716rexbii 3162 . . . . . . . 8 (∃𝑙 ∈ {∅}∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
18 0ex 5176 . . . . . . . . . . 11 ∅ ∈ V
19 breq2 5035 . . . . . . . . . . . 12 (𝑟 = ∅ → (𝑙 <<s 𝑟𝑙 <<s ∅))
20 oveq2 7181 . . . . . . . . . . . . 13 (𝑟 = ∅ → (𝑙 |s 𝑟) = (𝑙 |s ∅))
2120eqeq1d 2741 . . . . . . . . . . . 12 (𝑟 = ∅ → ((𝑙 |s 𝑟) = 𝑥 ↔ (𝑙 |s ∅) = 𝑥))
2219, 21anbi12d 634 . . . . . . . . . . 11 (𝑟 = ∅ → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥)))
2318, 22rexsn 4574 . . . . . . . . . 10 (∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
2423rexbii 3162 . . . . . . . . 9 (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
25 breq1 5034 . . . . . . . . . . 11 (𝑙 = ∅ → (𝑙 <<s ∅ ↔ ∅ <<s ∅))
26 oveq1 7180 . . . . . . . . . . . 12 (𝑙 = ∅ → (𝑙 |s ∅) = (∅ |s ∅))
2726eqeq1d 2741 . . . . . . . . . . 11 (𝑙 = ∅ → ((𝑙 |s ∅) = 𝑥 ↔ (∅ |s ∅) = 𝑥))
2825, 27anbi12d 634 . . . . . . . . . 10 (𝑙 = ∅ → ((𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥)))
2918, 28rexsn 4574 . . . . . . . . 9 (∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
3024, 29bitri 278 . . . . . . . 8 (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
3115, 17, 303bitri 300 . . . . . . 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 33664 . . . . . . 7 0s = (∅ |s ∅)
3433eqeq1i 2744 . . . . . 6 ( 0s = 𝑥 ↔ (∅ |s ∅) = 𝑥)
35 eqcom 2746 . . . . . 6 ( 0s = 𝑥𝑥 = 0s )
3632, 34, 353bitr2i 302 . . . . 5 (∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ 𝑥 = 0s )
3736anbi2i 626 . . . 4 ((𝑥 No ∧ ∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ (𝑥 No 𝑥 = 0s ))
38 0sno 33666 . . . . . 6 0s ∈ No
39 eleq1 2821 . . . . . 6 (𝑥 = 0s → (𝑥 No ↔ 0s ∈ No ))
4038, 39mpbiri 261 . . . . 5 (𝑥 = 0s → 𝑥 No )
4140pm4.71ri 564 . . . 4 (𝑥 = 0s ↔ (𝑥 No 𝑥 = 0s ))
4237, 41bitr4i 281 . . 3 ((𝑥 No ∧ ∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ 𝑥 = 0s )
434, 42mpgbir 1806 . 2 {𝑥 No ∣ ∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)} = { 0s }
443, 43eqtri 2762 1 ( M ‘∅) = { 0s }
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1542  wcel 2114  wrex 3055  {crab 3058  c0 4212  𝒫 cpw 4489  {csn 4517   cuni 4797   class class class wbr 5031  cima 5529  Oncon0 6173  cfv 6340  (class class class)co 7173   No csur 33489   <<s csslt 33621   |s cscut 33623   0s c0s 33662   M cmade 33672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-int 4838  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7130  df-ov 7176  df-oprab 7177  df-mpo 7178  df-wrecs 7979  df-recs 8040  df-1o 8134  df-2o 8135  df-no 33492  df-slt 33493  df-bday 33494  df-sslt 33622  df-scut 33624  df-0s 33664  df-made 33677
This theorem is referenced by:  new0  33704
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