Theorem made0 27357

Description: The only surreal made on day ∅ is 0_s. (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.

Step	Hyp	Ref	Expression
1		0elon 6415	. . 3 ⊢ ∅ ∈ On
2		madeval2 27337	. . 3 ⊢ (∅ ∈ On → ( M ‘∅) = {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)})
3	1, 2	ax-mp 5	. 2 ⊢ ( M ‘∅) = {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)}
4		rabeqsn 4668	. . 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 5353	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 No
6		nulssgt 27288	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ ∅ <<s ∅
8		ima0 6073	. . . . . . . . . . . . 13 ⊢ ( M “ ∅) = ∅
9	8	unieqi 4920	. . . . . . . . . . . 12 ⊢ ∪ ( M “ ∅) = ∪ ∅
10		uni0 4938	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
11	9, 10	eqtri 2760	. . . . . . . . . . 11 ⊢ ∪ ( M “ ∅) = ∅
12	11	pweqi 4617	. . . . . . . . . 10 ⊢ 𝒫 ∪ ( M “ ∅) = 𝒫 ∅
13		pw0 4814	. . . . . . . . . 10 ⊢ 𝒫 ∅ = {∅}
14	12, 13	eqtri 2760	. . . . . . . . 9 ⊢ 𝒫 ∪ ( M “ ∅) = {∅}
15	14	rexeqi 3324	. . . . . . . 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))
16	14	rexeqi 3324	. . . . . . . . 9 ⊢ (∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
17	16	rexbii 3094	. . . . . . . 8 ⊢ (∃𝑙 ∈ {∅}∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
18		0ex 5306	. . . . . . . . . . 11 ⊢ ∅ ∈ V
19		breq2 5151	. . . . . . . . . . . 12 ⊢ (𝑟 = ∅ → (𝑙 <<s 𝑟 ↔ 𝑙 <<s ∅))
20		oveq2 7413	. . . . . . . . . . . . 13 ⊢ (𝑟 = ∅ → (𝑙 |s 𝑟) = (𝑙 |s ∅))
21	20	eqeq1d 2734	. . . . . . . . . . . 12 ⊢ (𝑟 = ∅ → ((𝑙 |s 𝑟) = 𝑥 ↔ (𝑙 |s ∅) = 𝑥))
22	19, 21	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑟 = ∅ → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥)))
23	18, 22	rexsn 4685	. . . . . . . . . 10 ⊢ (∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
24	23	rexbii 3094	. . . . . . . . 9 ⊢ (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
25		breq1 5150	. . . . . . . . . . 11 ⊢ (𝑙 = ∅ → (𝑙 <<s ∅ ↔ ∅ <<s ∅))
26		oveq1 7412	. . . . . . . . . . . 12 ⊢ (𝑙 = ∅ → (𝑙 |s ∅) = (∅ |s ∅))
27	26	eqeq1d 2734	. . . . . . . . . . 11 ⊢ (𝑙 = ∅ → ((𝑙 |s ∅) = 𝑥 ↔ (∅ |s ∅) = 𝑥))
28	25, 27	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑙 = ∅ → ((𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥)))
29	18, 28	rexsn 4685	. . . . . . . . 9 ⊢ (∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
30	24, 29	bitri 274	. . . . . . . 8 ⊢ (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
31	15, 17, 30	3bitri 296	. . . . . . 7 ⊢ (∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
32	7, 31	mpbiran 707	. . . . . 6 ⊢ (∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ |s ∅) = 𝑥)
33		df-0s 27314	. . . . . . 7 ⊢ 0s = (∅ |s ∅)
34	33	eqeq1i 2737	. . . . . 6 ⊢ ( 0s = 𝑥 ↔ (∅ |s ∅) = 𝑥)
35		eqcom 2739	. . . . . 6 ⊢ ( 0s = 𝑥 ↔ 𝑥 = 0s )
36	32, 34, 35	3bitr2i 298	. . . . 5 ⊢ (∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ 𝑥 = 0s )
37	36	anbi2i 623	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ (𝑥 ∈ No ∧ 𝑥 = 0s ))
38		0sno 27316	. . . . . 6 ⊢ 0s ∈ No
39		eleq1 2821	. . . . . 6 ⊢ (𝑥 = 0s → (𝑥 ∈ No ↔ 0s ∈ No ))
40	38, 39	mpbiri 257	. . . . 5 ⊢ (𝑥 = 0s → 𝑥 ∈ No )
41	40	pm4.71ri 561	. . . 4 ⊢ (𝑥 = 0s ↔ (𝑥 ∈ No ∧ 𝑥 = 0s ))
42	37, 41	bitr4i 277	. . 3 ⊢ ((𝑥 ∈ No ∧ ∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ 𝑥 = 0s )
43	4, 42	mpgbir 1801	. 2 ⊢ {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)} = { 0s }
44	3, 43	eqtri 2760	1 ⊢ ( M ‘∅) = { 0s }

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  {crab 3432  ∅c0 4321  𝒫 cpw 4601  {csn 4627  ∪ cuni 4907   class class class wbr 5147   “ cima 5678  Oncon0 6361  ‘cfv 6540  (class class class)co 7405   No csur 27132   <<s csslt 27271   |s cscut 27273   0_s c0s 27312   M cmade 27326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-1o 8462  df-2o 8463  df-no 27135  df-slt 27136  df-bday 27137  df-sslt 27272  df-scut 27274  df-0s 27314  df-made 27331

This theorem is referenced by:  new0  27358  old1  27359