Theorem made0 27933

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 6397	. . 3 ⊢ ∅ ∈ On
2		madeval2 27903	. . 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 4625	. . 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		nulsgts 27846	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ ∅ <<s ∅
8		ima0 6063	. . . . . . . . . . . . 13 ⊢ ( M “ ∅) = ∅
9	8	unieqi 4876	. . . . . . . . . . . 12 ⊢ ∪ ( M “ ∅) = ∪ ∅
10		uni0 4893	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
11	9, 10	eqtri 2784	. . . . . . . . . . 11 ⊢ ∪ ( M “ ∅) = ∅
12	11	pweqi 4570	. . . . . . . . . 10 ⊢ 𝒫 ∪ ( M “ ∅) = 𝒫 ∅
13		pw0 4769	. . . . . . . . . 10 ⊢ 𝒫 ∅ = {∅}
14	12, 13	eqtri 2784	. . . . . . . . 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 3108	. . . . . . . 8 ⊢ (∃𝑙 ∈ {∅}∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥))
18		0ex 5256	. . . . . . . . . . 11 ⊢ ∅ ∈ V
19		breq2 5103	. . . . . . . . . . . 12 ⊢ (𝑟 = ∅ → (𝑙 <<s 𝑟 ↔ 𝑙 <<s ∅))
20		oveq2 7400	. . . . . . . . . . . . 13 ⊢ (𝑟 = ∅ → (𝑙 |s 𝑟) = (𝑙 |s ∅))
21	20	eqeq1d 2763	. . . . . . . . . . . 12 ⊢ (𝑟 = ∅ → ((𝑙 |s 𝑟) = 𝑥 ↔ (𝑙 |s ∅) = 𝑥))
22	19, 21	anbi12d 641	. . . . . . . . . . 11 ⊢ (𝑟 = ∅ → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥)))
23	18, 22	rexsn 4640	. . . . . . . . . 10 ⊢ (∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
24	23	rexbii 3108	. . . . . . . . 9 ⊢ (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ ∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥))
25		breq1 5102	. . . . . . . . . . 11 ⊢ (𝑙 = ∅ → (𝑙 <<s ∅ ↔ ∅ <<s ∅))
26		oveq1 7399	. . . . . . . . . . . 12 ⊢ (𝑙 = ∅ → (𝑙 |s ∅) = (∅ |s ∅))
27	26	eqeq1d 2763	. . . . . . . . . . 11 ⊢ (𝑙 = ∅ → ((𝑙 |s ∅) = 𝑥 ↔ (∅ |s ∅) = 𝑥))
28	25, 27	anbi12d 641	. . . . . . . . . 10 ⊢ (𝑙 = ∅ → ((𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥)))
29	18, 28	rexsn 4640	. . . . . . . . 9 ⊢ (∃𝑙 ∈ {∅} (𝑙 <<s ∅ ∧ (𝑙 |s ∅) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
30	24, 29	bitri 277	. . . . . . . 8 ⊢ (∃𝑙 ∈ {∅}∃𝑟 ∈ {∅} (𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
31	15, 17, 30	3bitri 299	. . . . . . 7 ⊢ (∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ <<s ∅ ∧ (∅ |s ∅) = 𝑥))
32	7, 31	mpbiran 719	. . . . . 6 ⊢ (∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ (∅ |s ∅) = 𝑥)
33		df-0s 27877	. . . . . . 7 ⊢ 0s = (∅ |s ∅)
34	33	eqeq1i 2766	. . . . . 6 ⊢ ( 0s = 𝑥 ↔ (∅ |s ∅) = 𝑥)
35		eqcom 2768	. . . . . 6 ⊢ ( 0s = 𝑥 ↔ 𝑥 = 0s )
36	32, 34, 35	3bitr2i 301	. . . . 5 ⊢ (∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) ↔ 𝑥 = 0s )
37	36	anbi2i 632	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ (𝑥 ∈ No ∧ 𝑥 = 0s ))
38		0no 27879	. . . . . 6 ⊢ 0s ∈ No
39		eleq1 2849	. . . . . 6 ⊢ (𝑥 = 0s → (𝑥 ∈ No ↔ 0s ∈ No ))
40	38, 39	mpbiri 260	. . . . 5 ⊢ (𝑥 = 0s → 𝑥 ∈ No )
41	40	pm4.71ri 568	. . . 4 ⊢ (𝑥 = 0s ↔ (𝑥 ∈ No ∧ 𝑥 = 0s ))
42	37, 41	bitr4i 280	. . 3 ⊢ ((𝑥 ∈ No ∧ ∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)) ↔ 𝑥 = 0s )
43	4, 42	mpgbir 1818	. 2 ⊢ {𝑥 ∈ No ∣ ∃𝑙 ∈ 𝒫 ∪ ( M “ ∅)∃𝑟 ∈ 𝒫 ∪ ( M “ ∅)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)} = { 0s }
44	3, 43	eqtri 2784	1 ⊢ ( M ‘∅) = { 0s }

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  {crab 3413  ∅c0 4285  𝒫 cpw 4554  {csn 4581  ∪ cuni 4864   class class class wbr 5099   “ cima 5648  Oncon0 6342  ‘cfv 6517  (class class class)co 7392   No csur 27681   <<s cslts 27827   |s ccuts 27829   0_s c0s 27875   M cmade 27892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-1o 8432  df-2o 8433  df-no 27684  df-lts 27685  df-bday 27686  df-slts 27828  df-cuts 27830  df-0s 27877  df-made 27897

This theorem is referenced by:  new0  27934  old1  27935