Theorem made0 27930

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 6449	. . 3 ⊢ ∅ ∈ On
2		madeval2 27910	. . 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 4689	. . 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 5374	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 No
6		nulssgt 27861	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ ∅ <<s ∅
8		ima0 6106	. . . . . . . . . . . . 13 ⊢ ( M “ ∅) = ∅
9	8	unieqi 4943	. . . . . . . . . . . 12 ⊢ ∪ ( M “ ∅) = ∪ ∅
10		uni0 4959	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
11	9, 10	eqtri 2768	. . . . . . . . . . 11 ⊢ ∪ ( M “ ∅) = ∅
12	11	pweqi 4638	. . . . . . . . . 10 ⊢ 𝒫 ∪ ( M “ ∅) = 𝒫 ∅
13		pw0 4837	. . . . . . . . . 10 ⊢ 𝒫 ∅ = {∅}
14	12, 13	eqtri 2768	. . . . . . . . 9 ⊢ 𝒫 ∪ ( M “ ∅) = {∅}
15	14	rexeqi 3333	. . . . . . . 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 3333	. . . . . . . . 9 ⊢ (∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
17	16	rexbii 3100	. . . . . . . 8 ⊢ (∃𝑙 ∈ {∅}∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
18		0ex 5325	. . . . . . . . . . 11 ⊢ ∅ ∈ V
19		breq2 5170	. . . . . . . . . . . 12 ⊢ (𝑟 = ∅ → (𝑙 <<s 𝑟 ↔ 𝑙 <<s ∅))
20		oveq2 7456	. . . . . . . . . . . . 13 ⊢ (𝑟 = ∅ → (𝑙 |s 𝑟) = (𝑙 |s ∅))
21	20	eqeq1d 2742	. . . . . . . . . . . 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 4706	. . . . . . . . . 10 ⊢ (∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
24	23	rexbii 3100	. . . . . . . . 9 ⊢ (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
25		breq1 5169	. . . . . . . . . . 11 ⊢ (𝑙 = ∅ → (𝑙 <<s ∅ ↔ ∅ <<s ∅))
26		oveq1 7455	. . . . . . . . . . . 12 ⊢ (𝑙 = ∅ → (𝑙 |s ∅) = (∅ |s ∅))
27	26	eqeq1d 2742	. . . . . . . . . . 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 4706	. . . . . . . . 9 ⊢ (∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
30	24, 29	bitri 275	. . . . . . . 8 ⊢ (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
31	15, 17, 30	3bitri 297	. . . . . . 7 ⊢ (∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
32	7, 31	mpbiran 708	. . . . . 6 ⊢ (∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ |s ∅) = 𝑥)
33		df-0s 27887	. . . . . . 7 ⊢ 0s = (∅ |s ∅)
34	33	eqeq1i 2745	. . . . . 6 ⊢ ( 0s = 𝑥 ↔ (∅ |s ∅) = 𝑥)
35		eqcom 2747	. . . . . 6 ⊢ ( 0s = 𝑥 ↔ 𝑥 = 0s )
36	32, 34, 35	3bitr2i 299	. . . . 5 ⊢ (∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ 𝑥 = 0s )
37	36	anbi2i 622	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ (𝑥 ∈ No ∧ 𝑥 = 0s ))
38		0sno 27889	. . . . . 6 ⊢ 0s ∈ No
39		eleq1 2832	. . . . . 6 ⊢ (𝑥 = 0s → (𝑥 ∈ No ↔ 0s ∈ No ))
40	38, 39	mpbiri 258	. . . . 5 ⊢ (𝑥 = 0s → 𝑥 ∈ No )
41	40	pm4.71ri 560	. . . 4 ⊢ (𝑥 = 0s ↔ (𝑥 ∈ No ∧ 𝑥 = 0s ))
42	37, 41	bitr4i 278	. . 3 ⊢ ((𝑥 ∈ No ∧ ∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ 𝑥 = 0s )
43	4, 42	mpgbir 1797	. 2 ⊢ {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)} = { 0s }
44	3, 43	eqtri 2768	1 ⊢ ( M ‘∅) = { 0s }

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃wrex 3076  {crab 3443  ∅c0 4352  𝒫 cpw 4622  {csn 4648  ∪ cuni 4931   class class class wbr 5166   “ cima 5703  Oncon0 6395  ‘cfv 6573  (class class class)co 7448   No csur 27702   <<s csslt 27843   |s cscut 27845   0_s c0s 27885   M cmade 27899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-0s 27887  df-made 27904

This theorem is referenced by:  new0  27931  old1  27932