Theorem dfn0s2 28340

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 27817	. . . . . 6 ⊢ 0s ∈ No
2	1	elexi 3465	. . . . 5 ⊢ 0s ∈ V
3	2	elintab 4916	. . . 4 ⊢ ( 0s ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ∀𝑥(( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 0s ∈ 𝑥))
4		simpl 482	. . . 4 ⊢ (( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 0s ∈ 𝑥)
5	3, 4	mpgbir 1801	. . 3 ⊢ 0s ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
6		oveq1 7375	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 +s 1s ) = (𝑧 +s 1s ))
7	6	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝑦 +s 1s ) ∈ 𝑥 ↔ (𝑧 +s 1s ) ∈ 𝑥))
8	7	rspccv 3575	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥 → (𝑧 ∈ 𝑥 → (𝑧 +s 1s ) ∈ 𝑥))
9	8	adantl 481	. . . . . . 7 ⊢ (( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → (𝑧 ∈ 𝑥 → (𝑧 +s 1s ) ∈ 𝑥))
10	9	a2i 14	. . . . . 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 1813	. . . . 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 3446	. . . . . 6 ⊢ 𝑧 ∈ V
13	12	elintab 4916	. . . . 5 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ∀𝑥(( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 𝑧 ∈ 𝑥))
14		ovex 7401	. . . . . 6 ⊢ (𝑧 +s 1s ) ∈ V
15	14	elintab 4916	. . . . 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 292	. . . 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 3054	. . 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 28327	. . 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 693	. 2 ⊢ ℕ0s ⊆ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
20		0n0s 28337	. . . 4 ⊢ 0s ∈ ℕ0s
21		peano2n0s 28338	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
22	21	rgen 3054	. . . 4 ⊢ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s
23		n0sex 28325	. . . . 5 ⊢ ℕ0s ∈ V
24		eleq2 2826	. . . . . 6 ⊢ (𝑥 = ℕ0s → ( 0s ∈ 𝑥 ↔ 0s ∈ ℕ0s))
25		eleq2 2826	. . . . . . 7 ⊢ (𝑥 = ℕ0s → ((𝑦 +s 1s ) ∈ 𝑥 ↔ (𝑦 +s 1s ) ∈ ℕ0s))
26	25	raleqbi1dv 3310	. . . . . 6 ⊢ (𝑥 = ℕ0s → (∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥 ↔ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s))
27	24, 26	anbi12d 633	. . . . 5 ⊢ (𝑥 = ℕ0s → (( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) ↔ ( 0s ∈ ℕ0s ∧ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s)))
28	23, 27	elab 3636	. . . 4 ⊢ (ℕ0s ∈ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ( 0s ∈ ℕ0s ∧ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s))
29	20, 22, 28	mpbir2an 712	. . 3 ⊢ ℕ0s ∈ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
30		intss1 4920	. . 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 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052   ⊆ wss 3903  ∩ cint 4904  (class class class)co 7368   No csur 27619   0_s c0s 27813   1_s c1s 27814   +_s cadds 27967  ℕ_0scn0s 28320

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  ax-inf2 9562

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-iun 4950  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-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  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-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  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-n0s 28322

This theorem is referenced by: (None)