Theorem dfn0s2 28501

Description: Alternate definition of the set of non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
dfn0s2	⊢ ℕ0s = ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}

Distinct variable group:   𝑥,𝑦

Proof of Theorem dfn0s2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0no 27978	. . . . . 6 ⊢ 0s ∈ No
2	1	elexi 3475	. . . . 5 ⊢ 0s ∈ V
3	2	elintab 4923	. . . 4 ⊢ ( 0s ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ∀𝑥(( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 0s ∈ 𝑥))
4		simpl 487	. . . 4 ⊢ (( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 0s ∈ 𝑥)
5	3, 4	mpgbir 1827	. . 3 ⊢ 0s ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
6		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 +s 1s ) = (𝑧 +s 1s ))
7	6	eleq1d 2846	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝑦 +s 1s ) ∈ 𝑥 ↔ (𝑧 +s 1s ) ∈ 𝑥))
8	7	rspccv 3577	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥 → (𝑧 ∈ 𝑥 → (𝑧 +s 1s ) ∈ 𝑥))
9	8	adantl 486	. . . . . . 7 ⊢ (( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → (𝑧 ∈ 𝑥 → (𝑧 +s 1s ) ∈ 𝑥))
10	9	a2i 15	. . . . . 6 ⊢ ((( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 𝑧 ∈ 𝑥) → (( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → (𝑧 +s 1s ) ∈ 𝑥))
11	10	alimi 1839	. . . . 5 ⊢ (∀𝑥(( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 𝑧 ∈ 𝑥) → ∀𝑥(( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → (𝑧 +s 1s ) ∈ 𝑥))
12		vex 3457	. . . . . 6 ⊢ 𝑧 ∈ V
13	12	elintab 4923	. . . . 5 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ∀𝑥(( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 𝑧 ∈ 𝑥))
14		ovex 7443	. . . . . 6 ⊢ (𝑧 +s 1s ) ∈ V
15	14	elintab 4923	. . . . 5 ⊢ ((𝑧 +s 1s ) ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ∀𝑥(( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → (𝑧 +s 1s ) ∈ 𝑥))
16	11, 13, 15	3imtr4i 295	. . . 4 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} → (𝑧 +s 1s ) ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)})
17	16	rgen 3079	. . 3 ⊢ ∀𝑧 ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} (𝑧 +s 1s ) ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
18		peano5n0s 28488	. . 3 ⊢ (( 0s ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ∧ ∀𝑧 ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} (𝑧 +s 1s ) ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}) → ℕ0s ⊆ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)})
19	5, 17, 18	mp2an 704	. 2 ⊢ ℕ0s ⊆ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
20		0n0s 28498	. . . 4 ⊢ 0s ∈ ℕ0s
21		peano2n0s 28499	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
22	21	rgen 3079	. . . 4 ⊢ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s
23		n0sex 28486	. . . . 5 ⊢ ℕ0s ∈ V
24		eleq2 2850	. . . . . 6 ⊢ (𝑥 = ℕ0s → ( 0s ∈ 𝑥 ↔ 0s ∈ ℕ0s))
25		eleq2 2850	. . . . . . 7 ⊢ (𝑥 = ℕ0s → ((𝑦 +s 1s ) ∈ 𝑥 ↔ (𝑦 +s 1s ) ∈ ℕ0s))
26	25	raleqbi1dv 3331	. . . . . 6 ⊢ (𝑥 = ℕ0s → (∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥 ↔ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s))
27	24, 26	anbi12d 643	. . . . 5 ⊢ (𝑥 = ℕ0s → (( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) ↔ ( 0s ∈ ℕ0s ∧ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s)))
28	23, 27	elab 3637	. . . 4 ⊢ (ℕ0s ∈ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ( 0s ∈ ℕ0s ∧ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s))
29	20, 22, 28	mpbir2an 723	. . 3 ⊢ ℕ0s ∈ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
30		intss1 4927	. . 3 ⊢ (ℕ0s ∈ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} → ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ⊆ ℕ0s)
31	29, 30	ax-mp 5	. 2 ⊢ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ⊆ ℕ0s
32	19, 31	eqssi 3952	1 ⊢ ℕ0s = ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1566   = wceq 1568   ∈ wcel 2141  {cab 2739  ∀wral 3077   ⊆ wss 3904  ∩ cint 4911  (class class class)co 7410   No csur 27780   0_s c0s 27974   1_s c1s 27975   +_s cadds 28128  ℕ_0scn0s 28481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9609

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-no 27783  df-lts 27784  df-bday 27785  df-slts 27927  df-cuts 27929  df-0s 27976  df-n0s 28483

This theorem is referenced by: (None)