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Theorem dfn0s2 28337
Description: Alternate definition of the set of non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
Assertion
Ref Expression
dfn0s2 0s = {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
Distinct variable group:   𝑥,𝑦

Proof of Theorem dfn0s2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 0sno 27872 . . . . . 6 0s No
21elexi 3502 . . . . 5 0s ∈ V
32elintab 4957 . . . 4 ( 0s {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ∀𝑥(( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 0s𝑥))
4 simpl 482 . . . 4 (( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 0s𝑥)
53, 4mpgbir 1798 . . 3 0s {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
6 oveq1 7439 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑦 +s 1s ) = (𝑧 +s 1s ))
76eleq1d 2825 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑦 +s 1s ) ∈ 𝑥 ↔ (𝑧 +s 1s ) ∈ 𝑥))
87rspccv 3618 . . . . . . . 8 (∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥 → (𝑧𝑥 → (𝑧 +s 1s ) ∈ 𝑥))
98adantl 481 . . . . . . 7 (( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥) → (𝑧𝑥 → (𝑧 +s 1s ) ∈ 𝑥))
109a2i 14 . . . . . 6 ((( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 𝑧𝑥) → (( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥) → (𝑧 +s 1s ) ∈ 𝑥))
1110alimi 1810 . . . . 5 (∀𝑥(( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 𝑧𝑥) → ∀𝑥(( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥) → (𝑧 +s 1s ) ∈ 𝑥))
12 vex 3483 . . . . . 6 𝑧 ∈ V
1312elintab 4957 . . . . 5 (𝑧 {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ∀𝑥(( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 𝑧𝑥))
14 ovex 7465 . . . . . 6 (𝑧 +s 1s ) ∈ V
1514elintab 4957 . . . . 5 ((𝑧 +s 1s ) ∈ {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ∀𝑥(( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥) → (𝑧 +s 1s ) ∈ 𝑥))
1611, 13, 153imtr4i 292 . . . 4 (𝑧 {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)} → (𝑧 +s 1s ) ∈ {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)})
1716rgen 3062 . . 3 𝑧 {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)} (𝑧 +s 1s ) ∈ {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
18 peano5n0s 28325 . . 3 (( 0s {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ∧ ∀𝑧 {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)} (𝑧 +s 1s ) ∈ {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)}) → ℕ0s {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)})
195, 17, 18mp2an 692 . 2 0s {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
20 0n0s 28335 . . . 4 0s ∈ ℕ0s
21 peano2n0s 28336 . . . . 5 (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
2221rgen 3062 . . . 4 𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s
23 n0sex 28323 . . . . 5 0s ∈ V
24 eleq2 2829 . . . . . 6 (𝑥 = ℕ0s → ( 0s𝑥 ↔ 0s ∈ ℕ0s))
25 eleq2 2829 . . . . . . 7 (𝑥 = ℕ0s → ((𝑦 +s 1s ) ∈ 𝑥 ↔ (𝑦 +s 1s ) ∈ ℕ0s))
2625raleqbi1dv 3337 . . . . . 6 (𝑥 = ℕ0s → (∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥 ↔ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s))
2724, 26anbi12d 632 . . . . 5 (𝑥 = ℕ0s → (( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥) ↔ ( 0s ∈ ℕ0s ∧ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s)))
2823, 27elab 3678 . . . 4 (ℕ0s ∈ {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ( 0s ∈ ℕ0s ∧ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s))
2920, 22, 28mpbir2an 711 . . 3 0s ∈ {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
30 intss1 4962 . . 3 (ℕ0s ∈ {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)} → {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ⊆ ℕ0s)
3129, 30ax-mp 5 . 2 {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ⊆ ℕ0s
3219, 31eqssi 3999 1 0s = {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1537   = wceq 1539  wcel 2107  {cab 2713  wral 3060  wss 3950   cint 4945  (class class class)co 7432   No csur 27685   0s c0s 27868   1s c1s 27869   +s cadds 27993  0scnn0s 28319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-dc 10487
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-no 27688  df-slt 27689  df-bday 27690  df-sslt 27827  df-scut 27829  df-0s 27870  df-n0s 28321
This theorem is referenced by: (None)
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