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Theorem dfn0s2 28351
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 27886 . . . . . 6 0s No
21elexi 3501 . . . . 5 0s ∈ V
32elintab 4963 . . . 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 1796 . . 3 0s {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
6 oveq1 7438 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑦 +s 1s ) = (𝑧 +s 1s ))
76eleq1d 2824 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑦 +s 1s ) ∈ 𝑥 ↔ (𝑧 +s 1s ) ∈ 𝑥))
87rspccv 3619 . . . . . . . 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 1808 . . . . 5 (∀𝑥(( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 𝑧𝑥) → ∀𝑥(( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥) → (𝑧 +s 1s ) ∈ 𝑥))
12 vex 3482 . . . . . 6 𝑧 ∈ V
1312elintab 4963 . . . . 5 (𝑧 {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ∀𝑥(( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 𝑧𝑥))
14 ovex 7464 . . . . . 6 (𝑧 +s 1s ) ∈ V
1514elintab 4963 . . . . 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 3061 . . 3 𝑧 {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)} (𝑧 +s 1s ) ∈ {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
18 peano5n0s 28339 . . 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 28349 . . . 4 0s ∈ ℕ0s
21 peano2n0s 28350 . . . . 5 (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
2221rgen 3061 . . . 4 𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s
23 n0sex 28337 . . . . 5 0s ∈ V
24 eleq2 2828 . . . . . 6 (𝑥 = ℕ0s → ( 0s𝑥 ↔ 0s ∈ ℕ0s))
25 eleq2 2828 . . . . . . 7 (𝑥 = ℕ0s → ((𝑦 +s 1s ) ∈ 𝑥 ↔ (𝑦 +s 1s ) ∈ ℕ0s))
2625raleqbi1dv 3336 . . . . . 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 3681 . . . 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 4968 . . 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 4012 1 0s = {𝑥 ∣ ( 0s𝑥 ∧ ∀𝑦𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wss 3963   cint 4951  (class class class)co 7431   No csur 27699   0s c0s 27882   1s c1s 27883   +s cadds 28007  0scnn0s 28333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-dc 10484
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-0s 27884  df-n0s 28335
This theorem is referenced by: (None)
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