Theorem made0 27203

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 6371	. . 3 ⊢ ∅ ∈ On
2		madeval2 27183	. . 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 4627	. . 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 5311	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 No
6		nulssgt 27137	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ ∅ <<s ∅
8		ima0 6029	. . . . . . . . . . . . 13 ⊢ ( M “ ∅) = ∅
9	8	unieqi 4878	. . . . . . . . . . . 12 ⊢ ∪ ( M “ ∅) = ∪ ∅
10		uni0 4896	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
11	9, 10	eqtri 2764	. . . . . . . . . . 11 ⊢ ∪ ( M “ ∅) = ∅
12	11	pweqi 4576	. . . . . . . . . 10 ⊢ 𝒫 ∪ ( M “ ∅) = 𝒫 ∅
13		pw0 4772	. . . . . . . . . 10 ⊢ 𝒫 ∅ = {∅}
14	12, 13	eqtri 2764	. . . . . . . . 9 ⊢ 𝒫 ∪ ( M “ ∅) = {∅}
15	14	rexeqi 3312	. . . . . . . 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 3312	. . . . . . . . 9 ⊢ (∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
17	16	rexbii 3097	. . . . . . . 8 ⊢ (∃𝑙 ∈ {∅}∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
18		0ex 5264	. . . . . . . . . . 11 ⊢ ∅ ∈ V
19		breq2 5109	. . . . . . . . . . . 12 ⊢ (𝑟 = ∅ → (𝑙 <<s 𝑟 ↔ 𝑙 <<s ∅))
20		oveq2 7365	. . . . . . . . . . . . 13 ⊢ (𝑟 = ∅ → (𝑙 |s 𝑟) = (𝑙 |s ∅))
21	20	eqeq1d 2738	. . . . . . . . . . . 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 4643	. . . . . . . . . 10 ⊢ (∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
24	23	rexbii 3097	. . . . . . . . 9 ⊢ (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
25		breq1 5108	. . . . . . . . . . 11 ⊢ (𝑙 = ∅ → (𝑙 <<s ∅ ↔ ∅ <<s ∅))
26		oveq1 7364	. . . . . . . . . . . 12 ⊢ (𝑙 = ∅ → (𝑙 |s ∅) = (∅ |s ∅))
27	26	eqeq1d 2738	. . . . . . . . . . 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 4643	. . . . . . . . 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 27163	. . . . . . 7 ⊢ 0s = (∅ |s ∅)
34	33	eqeq1i 2741	. . . . . 6 ⊢ ( 0s = 𝑥 ↔ (∅ |s ∅) = 𝑥)
35		eqcom 2743	. . . . . 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 27165	. . . . . 6 ⊢ 0s ∈ No
39		eleq1 2825	. . . . . 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 2764	1 ⊢ ( M ‘∅) = { 0s }

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3073  {crab 3407  ∅c0 4282  𝒫 cpw 4560  {csn 4586  ∪ cuni 4865   class class class wbr 5105   “ cima 5636  Oncon0 6317  ‘cfv 6496  (class class class)co 7357   No csur 26988   <<s csslt 27120   |s cscut 27122   0_s c0s 27161   M cmade 27172

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-1o 8412  df-2o 8413  df-no 26991  df-slt 26992  df-bday 26993  df-sslt 27121  df-scut 27123  df-0s 27163  df-made 27177

This theorem is referenced by:  new0  27204  old1  27205