Theorem eldifsucnn 8579

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 7820	. . . . . 6 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2		nnawordex 8552	. . . . . 6 ⊢ ((suc 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ⊆ 𝐵 ↔ ∃𝑦 ∈ ω (suc 𝐴 +o 𝑦) = 𝐵))
3	1, 2	sylan 580	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ⊆ 𝐵 ↔ ∃𝑦 ∈ ω (suc 𝐴 +o 𝑦) = 𝐵))
4		nnacl 8526	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) ∈ ω)
5		nnaword1 8544	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑦))
6		nnasuc 8521	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +o suc 𝐴) = suc (𝑦 +o 𝐴))
7	6	ancoms 458	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +o suc 𝐴) = suc (𝑦 +o 𝐴))
8		nnacom 8532	. . . . . . . . . . 11 ⊢ ((suc 𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 +o 𝑦) = (𝑦 +o suc 𝐴))
9	1, 8	sylan 580	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 +o 𝑦) = (𝑦 +o suc 𝐴))
10		nnacom 8532	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) = (𝑦 +o 𝐴))
11		suceq 6374	. . . . . . . . . . 11 ⊢ ((𝐴 +o 𝑦) = (𝑦 +o 𝐴) → suc (𝐴 +o 𝑦) = suc (𝑦 +o 𝐴))
12	10, 11	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝐴 +o 𝑦) = suc (𝑦 +o 𝐴))
13	7, 9, 12	3eqtr4d 2776	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 +o 𝑦) = suc (𝐴 +o 𝑦))
14		sseq2 3956	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 +o 𝑦) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (𝐴 +o 𝑦)))
15		suceq 6374	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐴 +o 𝑦) → suc 𝑥 = suc (𝐴 +o 𝑦))
16	15	eqeq2d 2742	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 +o 𝑦) → ((suc 𝐴 +o 𝑦) = suc 𝑥 ↔ (suc 𝐴 +o 𝑦) = suc (𝐴 +o 𝑦)))
17	14, 16	anbi12d 632	. . . . . . . . . 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 3572	. . . . . . . . 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 836	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ (suc 𝐴 +o 𝑦) = suc 𝑥))
20		eqeq1 2735	. . . . . . . . . 10 ⊢ ((suc 𝐴 +o 𝑦) = 𝐵 → ((suc 𝐴 +o 𝑦) = suc 𝑥 ↔ 𝐵 = suc 𝑥))
21	20	anbi2d 630	. . . . . . . . 9 ⊢ ((suc 𝐴 +o 𝑦) = 𝐵 → ((𝐴 ⊆ 𝑥 ∧ (suc 𝐴 +o 𝑦) = suc 𝑥) ↔ (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
22	21	rexbidv 3156	. . . . . . . 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 245	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((suc 𝐴 +o 𝑦) = 𝐵 → ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
24	23	rexlimdva 3133	. . . . . 6 ⊢ (𝐴 ∈ ω → (∃𝑦 ∈ ω (suc 𝐴 +o 𝑦) = 𝐵 → ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
25	24	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑦 ∈ ω (suc 𝐴 +o 𝑦) = 𝐵 → ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
26	3, 25	sylbid 240	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ⊆ 𝐵 → ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
27	26	expimpd 453	. . 3 ⊢ (𝐴 ∈ ω → ((𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵) → ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
28		peano2 7820	. . . . . . . 8 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
29	28	ad2antlr 727	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ⊆ 𝑥) → suc 𝑥 ∈ ω)
30		nnord 7804	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → Ord 𝐴)
31		nnord 7804	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → Ord 𝑥)
32		ordsucsssuc 7753	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ Ord 𝑥) → (𝐴 ⊆ 𝑥 ↔ suc 𝐴 ⊆ suc 𝑥))
33	30, 31, 32	syl2an 596	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ 𝑥 ↔ suc 𝐴 ⊆ suc 𝑥))
34	33	biimpa 476	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ⊆ 𝑥) → suc 𝐴 ⊆ suc 𝑥)
35	29, 34	jca 511	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ⊆ 𝑥) → (suc 𝑥 ∈ ω ∧ suc 𝐴 ⊆ suc 𝑥))
36		eleq1 2819	. . . . . . 7 ⊢ (𝐵 = suc 𝑥 → (𝐵 ∈ ω ↔ suc 𝑥 ∈ ω))
37		sseq2 3956	. . . . . . 7 ⊢ (𝐵 = suc 𝑥 → (suc 𝐴 ⊆ 𝐵 ↔ suc 𝐴 ⊆ suc 𝑥))
38	36, 37	anbi12d 632	. . . . . 6 ⊢ (𝐵 = suc 𝑥 → ((𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵) ↔ (suc 𝑥 ∈ ω ∧ suc 𝐴 ⊆ suc 𝑥)))
39	35, 38	syl5ibrcom 247	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ⊆ 𝑥) → (𝐵 = suc 𝑥 → (𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵)))
40	39	expimpd 453	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥) → (𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵)))
41	40	rexlimdva 3133	. . 3 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥) → (𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵)))
42	27, 41	impbid 212	. 2 ⊢ (𝐴 ∈ ω → ((𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵) ↔ ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
43		eldif 3907	. . 3 ⊢ (𝐵 ∈ (ω ∖ suc 𝐴) ↔ (𝐵 ∈ ω ∧ ¬ 𝐵 ∈ suc 𝐴))
44		nnord 7804	. . . . . 6 ⊢ (suc 𝐴 ∈ ω → Ord suc 𝐴)
45	1, 44	syl 17	. . . . 5 ⊢ (𝐴 ∈ ω → Ord suc 𝐴)
46		nnord 7804	. . . . 5 ⊢ (𝐵 ∈ ω → Ord 𝐵)
47		ordtri1 6339	. . . . 5 ⊢ ((Ord suc 𝐴 ∧ Ord 𝐵) → (suc 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ suc 𝐴))
48	45, 46, 47	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ suc 𝐴))
49	48	pm5.32da 579	. . 3 ⊢ (𝐴 ∈ ω → ((𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵) ↔ (𝐵 ∈ ω ∧ ¬ 𝐵 ∈ suc 𝐴)))
50	43, 49	bitr4id 290	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ (ω ∖ suc 𝐴) ↔ (𝐵 ∈ ω ∧ suc 𝐴 ⊆ 𝐵)))
51		eldif 3907	. . . . . 6 ⊢ (𝑥 ∈ (ω ∖ 𝐴) ↔ (𝑥 ∈ ω ∧ ¬ 𝑥 ∈ 𝐴))
52	51	anbi1i 624	. . . . 5 ⊢ ((𝑥 ∈ (ω ∖ 𝐴) ∧ 𝐵 = suc 𝑥) ↔ ((𝑥 ∈ ω ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝐵 = suc 𝑥))
53		anass 468	. . . . 5 ⊢ (((𝑥 ∈ ω ∧ ¬ 𝑥 ∈ 𝐴) ∧ 𝐵 = suc 𝑥) ↔ (𝑥 ∈ ω ∧ (¬ 𝑥 ∈ 𝐴 ∧ 𝐵 = suc 𝑥)))
54	52, 53	bitri 275	. . . 4 ⊢ ((𝑥 ∈ (ω ∖ 𝐴) ∧ 𝐵 = suc 𝑥) ↔ (𝑥 ∈ ω ∧ (¬ 𝑥 ∈ 𝐴 ∧ 𝐵 = suc 𝑥)))
55	54	rexbii2 3075	. . 3 ⊢ (∃𝑥 ∈ (ω ∖ 𝐴)𝐵 = suc 𝑥 ↔ ∃𝑥 ∈ ω (¬ 𝑥 ∈ 𝐴 ∧ 𝐵 = suc 𝑥))
56		ordtri1 6339	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝑥) → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴))
57	30, 31, 56	syl2an 596	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴))
58	57	anbi1d 631	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥) ↔ (¬ 𝑥 ∈ 𝐴 ∧ 𝐵 = suc 𝑥)))
59	58	rexbidva 3154	. . 3 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥) ↔ ∃𝑥 ∈ ω (¬ 𝑥 ∈ 𝐴 ∧ 𝐵 = suc 𝑥)))
60	55, 59	bitr4id 290	. 2 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ (ω ∖ 𝐴)𝐵 = suc 𝑥 ↔ ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
61	42, 50, 60	3bitr4d 311	1 ⊢ (𝐴 ∈ ω → (𝐵 ∈ (ω ∖ suc 𝐴) ↔ ∃𝑥 ∈ (ω ∖ 𝐴)𝐵 = suc 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056   ∖ cdif 3894   ⊆ wss 3897  Ord word 6305  suc csuc 6308  (class class class)co 7346  ωcom 7796   +_o coa 8382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-oadd 8389

This theorem is referenced by:  brttrcl2  9604  ttrcltr  9606  rnttrcl  9612