MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  made0 Structured version   Visualization version   GIF version

Theorem made0 27225
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 6376 . . 3 ∅ ∈ On
2 madeval2 27205 . . 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 4632 . . 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 5316 . . . . . . . 8 ∅ ∈ 𝒫 No
6 nulssgt 27159 . . . . . . . 8 (∅ ∈ 𝒫 No → ∅ <<s ∅)
75, 6ax-mp 5 . . . . . . 7 ∅ <<s ∅
8 ima0 6034 . . . . . . . . . . . . 13 ( M “ ∅) = ∅
98unieqi 4883 . . . . . . . . . . . 12 ( M “ ∅) =
10 uni0 4901 . . . . . . . . . . . 12 ∅ = ∅
119, 10eqtri 2765 . . . . . . . . . . 11 ( M “ ∅) = ∅
1211pweqi 4581 . . . . . . . . . 10 𝒫 ( M “ ∅) = 𝒫 ∅
13 pw0 4777 . . . . . . . . . 10 𝒫 ∅ = {∅}
1412, 13eqtri 2765 . . . . . . . . 9 𝒫 ( M “ ∅) = {∅}
1514rexeqi 3315 . . . . . . . 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 3315 . . . . . . . . 9 (∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
1716rexbii 3098 . . . . . . . 8 (∃𝑙 ∈ {∅}∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
18 0ex 5269 . . . . . . . . . . 11 ∅ ∈ V
19 breq2 5114 . . . . . . . . . . . 12 (𝑟 = ∅ → (𝑙 <<s 𝑟𝑙 <<s ∅))
20 oveq2 7370 . . . . . . . . . . . . 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 4648 . . . . . . . . . 10 (∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
2423rexbii 3098 . . . . . . . . 9 (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
25 breq1 5113 . . . . . . . . . . 11 (𝑙 = ∅ → (𝑙 <<s ∅ ↔ ∅ <<s ∅))
26 oveq1 7369 . . . . . . . . . . . 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 4648 . . . . . . . . 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 708 . . . . . 6 (∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ |s ∅) = 𝑥)
33 df-0s 27185 . . . . . . 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 624 . . . 4 ((𝑥 No ∧ ∃𝑙 ∈ 𝒫 ( M “ ∅)∃𝑟 ∈ 𝒫 ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ (𝑥 No 𝑥 = 0s ))
38 0sno 27187 . . . . . 6 0s No
39 eleq1 2826 . . . . . 6 (𝑥 = 0s → (𝑥 No ↔ 0s No ))
4038, 39mpbiri 258 . . . . 5 (𝑥 = 0s𝑥 No )
4140pm4.71ri 562 . . . 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 1802 . 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 205  wa 397   = wceq 1542  wcel 2107  wrex 3074  {crab 3410  c0 4287  𝒫 cpw 4565  {csn 4591   cuni 4870   class class class wbr 5110  cima 5641  Oncon0 6322  cfv 6501  (class class class)co 7362   No csur 27004   <<s csslt 27142   |s cscut 27144   0s c0s 27183   M cmade 27194
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-1o 8417  df-2o 8418  df-no 27007  df-slt 27008  df-bday 27009  df-sslt 27143  df-scut 27145  df-0s 27185  df-made 27199
This theorem is referenced by:  new0  27226  old1  27227
  Copyright terms: Public domain W3C validator