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Theorem bday1s 33668
Description: The birthday of surreal one is ordinal one. (Contributed by Scott Fenton, 8-Aug-2024.)
Assertion
Ref Expression
bday1s ( bday ‘ 1s ) = 1o

Proof of Theorem bday1s
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-1s 33662 . . 3 1s = ({ 0s } |s ∅)
21fveq2i 6677 . 2 ( bday ‘ 1s ) = ( bday ‘({ 0s } |s ∅))
3 0sno 33663 . . . . . . 7 0s ∈ No
4 snelpwi 5303 . . . . . . 7 ( 0s ∈ No → { 0s } ∈ 𝒫 No )
53, 4ax-mp 5 . . . . . 6 { 0s } ∈ 𝒫 No
6 nulssgt 33635 . . . . . 6 ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
75, 6ax-mp 5 . . . . 5 { 0s } <<s ∅
8 scutbdaybnd2 33653 . . . . 5 ({ 0s } <<s ∅ → ( bday ‘({ 0s } |s ∅)) ⊆ suc ( bday “ ({ 0s } ∪ ∅)))
97, 8ax-mp 5 . . . 4 ( bday ‘({ 0s } |s ∅)) ⊆ suc ( bday “ ({ 0s } ∪ ∅))
10 un0 4279 . . . . . . . . . 10 ({ 0s } ∪ ∅) = { 0s }
1110imaeq2i 5901 . . . . . . . . 9 ( bday “ ({ 0s } ∪ ∅)) = ( bday “ { 0s })
12 bdayfn 33611 . . . . . . . . . 10 bday Fn No
13 fnsnfv 6747 . . . . . . . . . 10 (( bday Fn No ∧ 0s ∈ No ) → {( bday ‘ 0s )} = ( bday “ { 0s }))
1412, 3, 13mp2an 692 . . . . . . . . 9 {( bday ‘ 0s )} = ( bday “ { 0s })
15 bday0s 33665 . . . . . . . . . 10 ( bday ‘ 0s ) = ∅
1615sneqi 4527 . . . . . . . . 9 {( bday ‘ 0s )} = {∅}
1711, 14, 163eqtr2i 2767 . . . . . . . 8 ( bday “ ({ 0s } ∪ ∅)) = {∅}
1817unieqi 4809 . . . . . . 7 ( bday “ ({ 0s } ∪ ∅)) = {∅}
19 0ex 5175 . . . . . . . 8 ∅ ∈ V
2019unisn 4818 . . . . . . 7 {∅} = ∅
2118, 20eqtri 2761 . . . . . 6 ( bday “ ({ 0s } ∪ ∅)) = ∅
22 suceq 6237 . . . . . 6 ( ( bday “ ({ 0s } ∪ ∅)) = ∅ → suc ( bday “ ({ 0s } ∪ ∅)) = suc ∅)
2321, 22ax-mp 5 . . . . 5 suc ( bday “ ({ 0s } ∪ ∅)) = suc ∅
24 df-1o 8131 . . . . 5 1o = suc ∅
2523, 24eqtr4i 2764 . . . 4 suc ( bday “ ({ 0s } ∪ ∅)) = 1o
269, 25sseqtri 3913 . . 3 ( bday ‘({ 0s } |s ∅)) ⊆ 1o
27 ssrab2 3969 . . . . . 6 {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
28 fnssintima 33249 . . . . . 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)))
2912, 27, 28mp2an 692 . . . . 5 (1o ( bday “ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}) ↔ ∀𝑦 ∈ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}1o ⊆ ( bday 𝑦))
30 sneq 4526 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
3130breq2d 5042 . . . . . . . 8 (𝑥 = 𝑦 → ({ 0s } <<s {𝑥} ↔ { 0s } <<s {𝑦}))
3230breq1d 5040 . . . . . . . 8 (𝑥 = 𝑦 → ({𝑥} <<s ∅ ↔ {𝑦} <<s ∅))
3331, 32anbi12d 634 . . . . . . 7 (𝑥 = 𝑦 → (({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅) ↔ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
3433elrab 3588 . . . . . 6 (𝑦 ∈ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ↔ (𝑦 No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
35 sltirr 33592 . . . . . . . . . . . . 13 ( 0s ∈ No → ¬ 0s <s 0s )
363, 35ax-mp 5 . . . . . . . . . . . 12 ¬ 0s <s 0s
37 breq2 5034 . . . . . . . . . . . 12 (𝑦 = 0s → ( 0s <s 𝑦 ↔ 0s <s 0s ))
3836, 37mtbiri 330 . . . . . . . . . . 11 (𝑦 = 0s → ¬ 0s <s 𝑦)
3938necon2ai 2963 . . . . . . . . . 10 ( 0s <s 𝑦𝑦 ≠ 0s )
40 bday0b 33667 . . . . . . . . . . 11 (𝑦 No → (( bday 𝑦) = ∅ ↔ 𝑦 = 0s ))
4140necon3bid 2978 . . . . . . . . . 10 (𝑦 No → (( bday 𝑦) ≠ ∅ ↔ 𝑦 ≠ 0s ))
4239, 41syl5ibr 249 . . . . . . . . 9 (𝑦 No → ( 0s <s 𝑦 → ( bday 𝑦) ≠ ∅))
43 bdayelon 33614 . . . . . . . . . . 11 ( bday 𝑦) ∈ On
4443onordi 6277 . . . . . . . . . 10 Ord ( bday 𝑦)
45 ordge1n0 8154 . . . . . . . . . 10 (Ord ( bday 𝑦) → (1o ⊆ ( bday 𝑦) ↔ ( bday 𝑦) ≠ ∅))
4644, 45ax-mp 5 . . . . . . . . 9 (1o ⊆ ( bday 𝑦) ↔ ( bday 𝑦) ≠ ∅)
4742, 46syl6ibr 255 . . . . . . . 8 (𝑦 No → ( 0s <s 𝑦 → 1o ⊆ ( bday 𝑦)))
48 ssltsep 33628 . . . . . . . . 9 ({ 0s } <<s {𝑦} → ∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧)
49 vex 3402 . . . . . . . . . . . 12 𝑦 ∈ V
50 breq2 5034 . . . . . . . . . . . 12 (𝑧 = 𝑦 → (𝑥 <s 𝑧𝑥 <s 𝑦))
5149, 50ralsn 4572 . . . . . . . . . . 11 (∀𝑧 ∈ {𝑦}𝑥 <s 𝑧𝑥 <s 𝑦)
5251ralbii 3080 . . . . . . . . . 10 (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ ∀𝑥 ∈ { 0s }𝑥 <s 𝑦)
533elexi 3417 . . . . . . . . . . 11 0s ∈ V
54 breq1 5033 . . . . . . . . . . 11 (𝑥 = 0s → (𝑥 <s 𝑦 ↔ 0s <s 𝑦))
5553, 54ralsn 4572 . . . . . . . . . 10 (∀𝑥 ∈ { 0s }𝑥 <s 𝑦 ↔ 0s <s 𝑦)
5652, 55bitri 278 . . . . . . . . 9 (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ 0s <s 𝑦)
5748, 56sylib 221 . . . . . . . 8 ({ 0s } <<s {𝑦} → 0s <s 𝑦)
5847, 57impel 509 . . . . . . 7 ((𝑦 No ∧ { 0s } <<s {𝑦}) → 1o ⊆ ( bday 𝑦))
5958adantrr 717 . . . . . 6 ((𝑦 No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)) → 1o ⊆ ( bday 𝑦))
6034, 59sylbi 220 . . . . 5 (𝑦 ∈ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} → 1o ⊆ ( bday 𝑦))
6129, 60mprgbir 3068 . . . 4 1o ( bday “ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)})
62 scutbday 33641 . . . . 5 ({ 0s } <<s ∅ → ( bday ‘({ 0s } |s ∅)) = ( bday “ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}))
637, 62ax-mp 5 . . . 4 ( bday ‘({ 0s } |s ∅)) = ( bday “ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)})
6461, 63sseqtrri 3914 . . 3 1o ⊆ ( bday ‘({ 0s } |s ∅))
6526, 64eqssi 3893 . 2 ( bday ‘({ 0s } |s ∅)) = 1o
662, 65eqtri 2761 1 ( bday ‘ 1s ) = 1o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399   = wceq 1542  wcel 2114  wne 2934  wral 3053  {crab 3057  cun 3841  wss 3843  c0 4211  𝒫 cpw 4488  {csn 4516   cuni 4796   cint 4836   class class class wbr 5030  cima 5528  Ord word 6171  suc csuc 6174   Fn wfn 6334  cfv 6339  (class class class)co 7170  1oc1o 8124   No csur 33486   <s cslt 33487   bday cbday 33488   <<s csslt 33618   |s cscut 33620   0s c0s 33659   1s c1s 33660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1o 8131  df-2o 8132  df-no 33489  df-slt 33490  df-bday 33491  df-sslt 33619  df-scut 33621  df-0s 33661  df-1s 33662
This theorem is referenced by: (None)
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