Theorem made0 28254

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 6418	. . 3 ⊢ ∅ ∈ On
2		madeval2 28224	. . 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 4624	. . 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		nulsgts 28167	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ ∅ <<s ∅
8		ima0 6074	. . . . . . . . . . . . 13 ⊢ (M “ ∅) = ∅
9	8	unieqi 4878	. . . . . . . . . . . 12 ⊢ ∪ (M “ ∅) = ∪ ∅
10		uni0 4895	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
11	9, 10	eqtri 2783	. . . . . . . . . . 11 ⊢ ∪ (M “ ∅) = ∅
12	11	pweqi 4572	. . . . . . . . . 10 ⊢ 𝒫 ∪ (M “ ∅) = 𝒫 ∅
13		pw0 4829	. . . . . . . . . 10 ⊢ 𝒫 ∅ = {∅}
14	12, 13	eqtri 2783	. . . . . . . . 9 ⊢ 𝒫 ∪ (M “ ∅) = {∅}
15	14	rexeqi 3318	. . . . . . . 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 3318	. . . . . . . . 9 ⊢ (∃𝑟 ∈ 𝒫 ∪ (M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
17	16	rexbii 3109	. . . . . . . 8 ⊢ (∃𝑙 ∈ {∅}∃𝑟 ∈ 𝒫 ∪ (M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
18		0ex 5260	. . . . . . . . . . 11 ⊢ ∅ ∈ V
19		breq2 5106	. . . . . . . . . . . 12 ⊢ (𝑟 = ∅ → (𝑙 <<s 𝑟 ↔ 𝑙 <<s ∅))
20		oveq2 7428	. . . . . . . . . . . . 13 ⊢ (𝑟 = ∅ → (𝑙 |s 𝑟) = (𝑙 |s ∅))
21	20	eqeq1d 2762	. . . . . . . . . . . 12 ⊢ (𝑟 = ∅ → ((𝑙 |s 𝑟) = 𝑥 ↔ (𝑙 |s ∅) = 𝑥))
22	19, 21	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑟 = ∅ → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥)))
23	18, 22	rexsn 4639	. . . . . . . . . 10 ⊢ (∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
24	23	rexbii 3109	. . . . . . . . 9 ⊢ (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
25		breq1 5105	. . . . . . . . . . 11 ⊢ (𝑙 = ∅ → (𝑙 <<s ∅ ↔ ∅ <<s ∅))
26		oveq1 7427	. . . . . . . . . . . 12 ⊢ (𝑙 = ∅ → (𝑙 |s ∅) = (∅ |s ∅))
27	26	eqeq1d 2762	. . . . . . . . . . 11 ⊢ (𝑙 = ∅ → ((𝑙 |s ∅) = 𝑥 ↔ (∅ |s ∅) = 𝑥))
28	25, 27	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑙 = ∅ → ((𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥)))
29	18, 28	rexsn 4639	. . . . . . . . 9 ⊢ (∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
30	24, 29	bitri 278	. . . . . . . 8 ⊢ (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
31	15, 17, 30	3bitri 300	. . . . . . 7 ⊢ (∃𝑙 ∈ 𝒫 ∪ (M “ ∅)∃𝑟 ∈ 𝒫 ∪ (M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
32	7, 31	mpbiran 722	. . . . . 6 ⊢ (∃𝑙 ∈ 𝒫 ∪ (M “ ∅)∃𝑟 ∈ 𝒫 ∪ (M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ |s ∅) = 𝑥)
33		df-0s 28198	. . . . . . 7 ⊢ 0s = (∅ |s ∅)
34	33	eqeq1i 2765	. . . . . 6 ⊢ ( 0s = 𝑥 ↔ (∅ |s ∅) = 𝑥)
35		eqcom 2767	. . . . . 6 ⊢ ( 0s = 𝑥 ↔ 𝑥 = 0s )
36	32, 34, 35	3bitr2i 302	. . . . 5 ⊢ (∃𝑙 ∈ 𝒫 ∪ (M “ ∅)∃𝑟 ∈ 𝒫 ∪ (M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ 𝑥 = 0s )
37	36	anbi2i 635	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∃𝑙 ∈ 𝒫 ∪ (M “ ∅)∃𝑟 ∈ 𝒫 ∪ (M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ (𝑥 ∈ No ∧ 𝑥 = 0s ))
38		0no 28200	. . . . . 6 ⊢ 0s ∈ No
39		eleq1 2848	. . . . . 6 ⊢ (𝑥 = 0s → (𝑥 ∈ No ↔ 0s ∈ No))
40	38, 39	mpbiri 261	. . . . 5 ⊢ (𝑥 = 0s → 𝑥 ∈ No)
41	40	pm4.71ri 570	. . . 4 ⊢ (𝑥 = 0s ↔ (𝑥 ∈ No ∧ 𝑥 = 0s ))
42	37, 41	bitr4i 281	. . 3 ⊢ ((𝑥 ∈ No ∧ ∃𝑙 ∈ 𝒫 ∪ (M “ ∅)∃𝑟 ∈ 𝒫 ∪ (M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ 𝑥 = 0s )
43	4, 42	mpgbir 1831	. 2 ⊢ {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ (M “ ∅)∃𝑟 ∈ 𝒫 ∪ (M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)} = { 0s }
44	3, 43	eqtri 2783	1 ⊢ (M‘∅) = { 0s }

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  {crab 3412  ∅c0 4278  𝒫 cpw 4556  {csn 4584  ∪ cuni 4866   class class class wbr 5102   “ cima 5653  Oncon0 6362  ‘cfv 6538  (class class class)co 7420  Nocsur 28002   <<s cslts 28148   |s ccuts 28150   0_s c0s 28196  Mcmade 28213

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151  df-0s 28198  df-made 28218

This theorem is used by:  new0  28255  old1  28256