Theorem eldifsucnn 8652

Description: Condition for membership in the difference of ω and a nonzero finite ordinal. (Contributed by Scott Fenton, 24-Oct-2024.)

Assertion

Ref	Expression
eldifsucnn	⊢ (𝐴 ∈ ω → (𝐵 ∈ (ω ∖ suc 𝐴) ↔ ∃𝑥 ∈ (ω ∖ 𝐴)𝐵 = suc 𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eldifsucnn

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2 7885	. . . . . 6 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2		nnawordex 8625	. . . . . 6 ⊢ ((suc 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ⊆ 𝐵 ↔ ∃𝑦 ∈ ω (suc 𝐴 +o 𝑦) = 𝐵))
3	1, 2	sylan 592	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ⊆ 𝐵 ↔ ∃𝑦 ∈ ω (suc 𝐴 +o 𝑦) = 𝐵))
4		nnacl 8599	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) ∈ ω)
5		nnaword1 8617	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑦))
6		nnasuc 8594	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +o suc 𝐴) = suc (𝑦 +o 𝐴))
7	6	ancoms 464	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +o suc 𝐴) = suc (𝑦 +o 𝐴))
8		nnacom 8605	. . . . . . . . . . 11 ⊢ ((suc 𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 +o 𝑦) = (𝑦 +o suc 𝐴))
9	1, 8	sylan 592	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 +o 𝑦) = (𝑦 +o suc 𝐴))
10		nnacom 8605	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) = (𝑦 +o 𝐴))
11		suceq 6421	. . . . . . . . . . 11 ⊢ ((𝐴 +o 𝑦) = (𝑦 +o 𝐴) → suc (𝐴 +o 𝑦) = suc (𝑦 +o 𝐴))
12	10, 11	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝐴 +o 𝑦) = suc (𝑦 +o 𝐴))
13	7, 9, 12	3eqtr4d 2805	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 +o 𝑦) = suc (𝐴 +o 𝑦))
14		sseq2 3957	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 +o 𝑦) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (𝐴 +o 𝑦)))
15		suceq 6421	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐴 +o 𝑦) → suc 𝑥 = suc (𝐴 +o 𝑦))
16	15	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 +o 𝑦) → ((suc 𝐴 +o 𝑦) = suc 𝑥 ↔ (suc 𝐴 +o 𝑦) = suc (𝐴 +o 𝑦)))
17	14, 16	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 +o 𝑦) → ((𝐴 ⊆ 𝑥 ∧ (suc 𝐴 +o 𝑦) = suc 𝑥) ↔ (𝐴 ⊆ (𝐴 +o 𝑦) ∧ (suc 𝐴 +o 𝑦) = suc (𝐴 +o 𝑦))))
18	17	rspcev 3576	. . . . . . . . 9 ⊢ (((𝐴 +o 𝑦) ∈ ω ∧ (𝐴 ⊆ (𝐴 +o 𝑦) ∧ (suc 𝐴 +o 𝑦) = suc (𝐴 +o 𝑦))) → ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ (suc 𝐴 +o 𝑦) = suc 𝑥))
19	4, 5, 13, 18	syl12anc 850	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ (suc 𝐴 +o 𝑦) = suc 𝑥))
20		eqeq1 2764	. . . . . . . . . 10 ⊢ ((suc 𝐴 +o 𝑦) = 𝐵 → ((suc 𝐴 +o 𝑦) = suc 𝑥 ↔ 𝐵 = suc 𝑥))
21	20	anbi2d 642	. . . . . . . . 9 ⊢ ((suc 𝐴 +o 𝑦) = 𝐵 → ((𝐴 ⊆ 𝑥 ∧ (suc 𝐴 +o 𝑦) = suc 𝑥) ↔ (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
22	21	rexbidv 3186	. . . . . . . 8 ⊢ ((suc 𝐴 +o 𝑦) = 𝐵 → (∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ (suc 𝐴 +o 𝑦) = suc 𝑥) ↔ ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
23	19, 22	syl5ibcom 248	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((suc 𝐴 +o 𝑦) = 𝐵 → ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
24	23	rexlimdva 3163	. . . . . 6 ⊢ (𝐴 ∈ ω → (∃𝑦 ∈ ω (suc 𝐴 +o 𝑦) = 𝐵 → ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
25	24	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑦 ∈ ω (suc 𝐴 +o 𝑦) = 𝐵 → ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
26	3, 25	sylbid 243	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ⊆ 𝐵 → ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
27	26	expimpd 459	. . 3 ⊢ (𝐴 ∈ ω → ((𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵) → ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
28		peano2 7885	. . . . . . . 8 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
29	28	ad2antlr 740	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ⊆ 𝑥) → suc 𝑥 ∈ ω)
30		nnord 7869	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → Ord 𝐴)
31		nnord 7869	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → Ord 𝑥)
32		ordsucsssuc 7818	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ Ord 𝑥) → (𝐴 ⊆ 𝑥 ↔ suc 𝐴 ⊆ suc 𝑥))
33	30, 31, 32	syl2an 608	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ 𝑥 ↔ suc 𝐴 ⊆ suc 𝑥))
34	33	biimpa 482	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ⊆ 𝑥) → suc 𝐴 ⊆ suc 𝑥)
35	29, 34	jca 521	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ⊆ 𝑥) → (suc 𝑥 ∈ ω ∧ suc 𝐴 ⊆ suc 𝑥))
36		eleq1 2848	. . . . . . 7 ⊢ (𝐵 = suc 𝑥 → (𝐵 ∈ ω ↔ suc 𝑥 ∈ ω))
37		sseq2 3957	. . . . . . 7 ⊢ (𝐵 = suc 𝑥 → (suc 𝐴 ⊆ 𝐵 ↔ suc 𝐴 ⊆ suc 𝑥))
38	36, 37	anbi12d 644	. . . . . 6 ⊢ (𝐵 = suc 𝑥 → ((𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵) ↔ (suc 𝑥 ∈ ω ∧ suc 𝐴 ⊆ suc 𝑥)))
39	35, 38	syl5ibrcom 250	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ⊆ 𝑥) → (𝐵 = suc 𝑥 → (𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵)))
40	39	expimpd 459	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥) → (𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵)))
41	40	rexlimdva 3163	. . 3 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥) → (𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵)))
42	27, 41	impbid 215	. 2 ⊢ (𝐴 ∈ ω → ((𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵) ↔ ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
43		eldif 3909	. . 3 ⊢ (𝐵 ∈ (ω ∖ suc 𝐴) ↔ (𝐵 ∈ ω ∧ ¬ 𝐵 ∈ suc 𝐴))
44		nnord 7869	. . . . . 6 ⊢ (suc 𝐴 ∈ ω → Ord suc 𝐴)
45	1, 44	syl 18	. . . . 5 ⊢ (𝐴 ∈ ω → Ord suc 𝐴)
46		nnord 7869	. . . . 5 ⊢ (𝐵 ∈ ω → Ord 𝐵)
47		ordtri1 6386	. . . . 5 ⊢ ((Ord suc 𝐴 ∧ Ord 𝐵) → (suc 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ suc 𝐴))
48	45, 46, 47	syl2an 608	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ suc 𝐴))
49	48	pm5.32da 590	. . 3 ⊢ (𝐴 ∈ ω → ((𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵) ↔ (𝐵 ∈ ω ∧ ¬ 𝐵 ∈ suc 𝐴)))
50	43, 49	bitr4id 293	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ (ω ∖ suc 𝐴) ↔ (𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵)))
51		eldif 3909	. . . . . 6 ⊢ (𝑥 ∈ (ω ∖ 𝐴) ↔ (𝑥 ∈ ω ∧ ¬ 𝑥 ∈ 𝐴))
52	51	anbi1i 636	. . . . 5 ⊢ ((𝑥 ∈ (ω ∖ 𝐴) ∧ 𝐵 = suc 𝑥) ↔ ((𝑥 ∈ ω ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝐵 = suc 𝑥))
53		anass 474	. . . . 5 ⊢ (((𝑥 ∈ ω ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝐵 = suc 𝑥) ↔ (𝑥 ∈ ω ∧ (¬ 𝑥 ∈ 𝐴 ∧ 𝐵 = suc 𝑥)))
54	52, 53	bitri 278	. . . 4 ⊢ ((𝑥 ∈ (ω ∖ 𝐴) ∧ 𝐵 = suc 𝑥) ↔ (𝑥 ∈ ω ∧ (¬ 𝑥 ∈ 𝐴 ∧ 𝐵 = suc 𝑥)))
55	54	rexbii2 3105	. . 3 ⊢ (∃𝑥 ∈ (ω ∖ 𝐴)𝐵 = suc 𝑥 ↔ ∃𝑥 ∈ ω (¬ 𝑥 ∈ 𝐴 ∧ 𝐵 = suc 𝑥))
56		ordtri1 6386	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝑥) → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴))
57	30, 31, 56	syl2an 608	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴))
58	57	anbi1d 643	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥) ↔ (¬ 𝑥 ∈ 𝐴 ∧ 𝐵 = suc 𝑥)))
59	58	rexbidva 3184	. . 3 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥) ↔ ∃𝑥 ∈ ω (¬ 𝑥 ∈ 𝐴 ∧ 𝐵 = suc 𝑥)))
60	55, 59	bitr4id 293	. 2 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ (ω ∖ 𝐴)𝐵 = suc 𝑥 ↔ ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
61	42, 50, 60	3bitr4d 314	1 ⊢ (𝐴 ∈ ω → (𝐵 ∈ (ω ∖ suc 𝐴) ↔ ∃𝑥 ∈ (ω ∖ 𝐴)𝐵 = suc 𝑥))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ∖ cdif 3896   ⊆ wss 3899  Ord word 6351  suc csuc 6354  (class class class)co 7409  ωcom 7861   +_o coa 8452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6294  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7862  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-oadd 8459

This theorem is used by:  brttrcl2  9693  ttrcltr  9695  rnttrcl  9701