Theorem bday1 27822

Description: The birthday of surreal one is ordinal one. (Contributed by Scott Fenton, 8-Aug-2024.)

Assertion

Ref	Expression
bday1	⊢ ( bday ‘ 1s ) = 1o

Proof of Theorem bday1

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-1s 27816	. . 3 ⊢ 1s = ({ 0s } |s ∅)
2	1	fveq2i 6845	. 2 ⊢ ( bday ‘ 1s ) = ( bday ‘({ 0s } |s ∅))
3		0no 27817	. . . . . . 7 ⊢ 0s ∈ No
4		snelpwi 5399	. . . . . . 7 ⊢ ( 0s ∈ No → { 0s } ∈ 𝒫 No )
5	3, 4	ax-mp 5	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
6		nulsgts 27784	. . . . . 6 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
7	5, 6	ax-mp 5	. . . . 5 ⊢ { 0s } <<s ∅
8		cutbdaybnd2 27804	. . . . 5 ⊢ ({ 0s } <<s ∅ → ( bday ‘({ 0s } |s ∅)) ⊆ suc ∪ ( bday “ ({ 0s } ∪ ∅)))
9	7, 8	ax-mp 5	. . . 4 ⊢ ( bday ‘({ 0s } |s ∅)) ⊆ suc ∪ ( bday “ ({ 0s } ∪ ∅))
10		un0 4348	. . . . . . . . . 10 ⊢ ({ 0s } ∪ ∅) = { 0s }
11	10	imaeq2i 6025	. . . . . . . . 9 ⊢ ( bday “ ({ 0s } ∪ ∅)) = ( bday “ { 0s })
12		bdayfn 27757	. . . . . . . . . 10 ⊢ bday Fn No
13		fnsnfv 6921	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 0s ∈ No ) → {( bday ‘ 0s )} = ( bday “ { 0s }))
14	12, 3, 13	mp2an 693	. . . . . . . . 9 ⊢ {( bday ‘ 0s )} = ( bday “ { 0s })
15		bday0 27819	. . . . . . . . . 10 ⊢ ( bday ‘ 0s ) = ∅
16	15	sneqi 4593	. . . . . . . . 9 ⊢ {( bday ‘ 0s )} = {∅}
17	11, 14, 16	3eqtr2i 2766	. . . . . . . 8 ⊢ ( bday “ ({ 0s } ∪ ∅)) = {∅}
18	17	unieqi 4877	. . . . . . 7 ⊢ ∪ ( bday “ ({ 0s } ∪ ∅)) = ∪ {∅}
19		0ex 5254	. . . . . . . 8 ⊢ ∅ ∈ V
20	19	unisn 4884	. . . . . . 7 ⊢ ∪ {∅} = ∅
21	18, 20	eqtri 2760	. . . . . 6 ⊢ ∪ ( bday “ ({ 0s } ∪ ∅)) = ∅
22		suceq 6393	. . . . . 6 ⊢ (∪ ( bday “ ({ 0s } ∪ ∅)) = ∅ → suc ∪ ( bday “ ({ 0s } ∪ ∅)) = suc ∅)
23	21, 22	ax-mp 5	. . . . 5 ⊢ suc ∪ ( bday “ ({ 0s } ∪ ∅)) = suc ∅
24		df-1o 8407	. . . . 5 ⊢ 1o = suc ∅
25	23, 24	eqtr4i 2763	. . . 4 ⊢ suc ∪ ( bday “ ({ 0s } ∪ ∅)) = 1o
26	9, 25	sseqtri 3984	. . 3 ⊢ ( bday ‘({ 0s } |s ∅)) ⊆ 1o
27		ssrab2 4034	. . . . . 6 ⊢ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
28		fnssintima 7318	. . . . . 6 ⊢ (( bday Fn No ∧ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No ) → (1o ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}) ↔ ∀𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}1o ⊆ ( bday ‘𝑦)))
29	12, 27, 28	mp2an 693	. . . . 5 ⊢ (1o ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}) ↔ ∀𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}1o ⊆ ( bday ‘𝑦))
30		sneq 4592	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
31	30	breq2d 5112	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ({ 0s } <<s {𝑥} ↔ { 0s } <<s {𝑦}))
32	30	breq1d 5110	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ({𝑥} <<s ∅ ↔ {𝑦} <<s ∅))
33	31, 32	anbi12d 633	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅) ↔ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
34	33	elrab 3648	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ↔ (𝑦 ∈ No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
35		ltsirr 27726	. . . . . . . . . . . . 13 ⊢ ( 0s ∈ No → ¬ 0s <s 0s )
36	3, 35	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 0s <s 0s
37		breq2 5104	. . . . . . . . . . . 12 ⊢ (𝑦 = 0s → ( 0s <s 𝑦 ↔ 0s <s 0s ))
38	36, 37	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → ¬ 0s <s 𝑦)
39	38	necon2ai 2962	. . . . . . . . . 10 ⊢ ( 0s <s 𝑦 → 𝑦 ≠ 0s )
40		bday0b 27821	. . . . . . . . . . 11 ⊢ (𝑦 ∈ No → (( bday ‘𝑦) = ∅ ↔ 𝑦 = 0s ))
41	40	necon3bid 2977	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → (( bday ‘𝑦) ≠ ∅ ↔ 𝑦 ≠ 0s ))
42	39, 41	imbitrrid 246	. . . . . . . . 9 ⊢ (𝑦 ∈ No → ( 0s <s 𝑦 → ( bday ‘𝑦) ≠ ∅))
43		bdayon 27760	. . . . . . . . . . 11 ⊢ ( bday ‘𝑦) ∈ On
44	43	onordi 6438	. . . . . . . . . 10 ⊢ Ord ( bday ‘𝑦)
45		ordge1n0 8431	. . . . . . . . . 10 ⊢ (Ord ( bday ‘𝑦) → (1o ⊆ ( bday ‘𝑦) ↔ ( bday ‘𝑦) ≠ ∅))
46	44, 45	ax-mp 5	. . . . . . . . 9 ⊢ (1o ⊆ ( bday ‘𝑦) ↔ ( bday ‘𝑦) ≠ ∅)
47	42, 46	imbitrrdi 252	. . . . . . . 8 ⊢ (𝑦 ∈ No → ( 0s <s 𝑦 → 1o ⊆ ( bday ‘𝑦)))
48		sltssep 27775	. . . . . . . . 9 ⊢ ({ 0s } <<s {𝑦} → ∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧)
49		vex 3446	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
50		breq2 5104	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑥 <s 𝑧 ↔ 𝑥 <s 𝑦))
51	49, 50	ralsn 4640	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ 𝑥 <s 𝑦)
52	51	ralbii 3084	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ ∀𝑥 ∈ { 0s }𝑥 <s 𝑦)
53	3	elexi 3465	. . . . . . . . . . 11 ⊢ 0s ∈ V
54		breq1 5103	. . . . . . . . . . 11 ⊢ (𝑥 = 0s → (𝑥 <s 𝑦 ↔ 0s <s 𝑦))
55	53, 54	ralsn 4640	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ { 0s }𝑥 <s 𝑦 ↔ 0s <s 𝑦)
56	52, 55	bitri 275	. . . . . . . . 9 ⊢ (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ 0s <s 𝑦)
57	48, 56	sylib 218	. . . . . . . 8 ⊢ ({ 0s } <<s {𝑦} → 0s <s 𝑦)
58	47, 57	impel 505	. . . . . . 7 ⊢ ((𝑦 ∈ No ∧ { 0s } <<s {𝑦}) → 1o ⊆ ( bday ‘𝑦))
59	58	adantrr 718	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)) → 1o ⊆ ( bday ‘𝑦))
60	34, 59	sylbi 217	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} → 1o ⊆ ( bday ‘𝑦))
61	29, 60	mprgbir 3059	. . . 4 ⊢ 1o ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)})
62		cutbday 27792	. . . . 5 ⊢ ({ 0s } <<s ∅ → ( bday ‘({ 0s } |s ∅)) = ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}))
63	7, 62	ax-mp 5	. . . 4 ⊢ ( bday ‘({ 0s } |s ∅)) = ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)})
64	61, 63	sseqtrri 3985	. . 3 ⊢ 1o ⊆ ( bday ‘({ 0s } |s ∅))
65	26, 64	eqssi 3952	. 2 ⊢ ( bday ‘({ 0s } |s ∅)) = 1o
66	2, 65	eqtri 2760	1 ⊢ ( bday ‘ 1s ) = 1o

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {crab 3401   ∪ cun 3901   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {csn 4582  ∪ cuni 4865  ∩ cint 4904   class class class wbr 5100   “ cima 5635  Ord word 6324  suc csuc 6327   Fn wfn 6495  ‘cfv 6500  (class class class)co 7368  1_oc1o 8400   No csur 27619   <s clts 27620   bday cbday 27621   <<s cslts 27765   |s ccuts 27767   0_s c0s 27813   1_s c1s 27814

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1o 8407  df-2o 8408  df-no 27622  df-lts 27623  df-bday 27624  df-slts 27766  df-cuts 27768  df-0s 27815  df-1s 27816

This theorem is referenced by:  cuteq1  27825  left1s  27903  right1s  27904  bdaypw2n0bndlem  28471