Theorem eldifsucnn 8604

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 7844	. . . . . 6 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2		nnawordex 8577	. . . . . 6 ⊢ ((suc 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ⊆ 𝐵 ↔ ∃𝑦 ∈ ω (suc 𝐴 +o 𝑦) = 𝐵))
3	1, 2	sylan 581	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ⊆ 𝐵 ↔ ∃𝑦 ∈ ω (suc 𝐴 +o 𝑦) = 𝐵))
4		nnacl 8551	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) ∈ ω)
5		nnaword1 8569	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑦))
6		nnasuc 8546	. . . . . . . . . . 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 8557	. . . . . . . . . . 11 ⊢ ((suc 𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 +o 𝑦) = (𝑦 +o suc 𝐴))
9	1, 8	sylan 581	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 +o 𝑦) = (𝑦 +o suc 𝐴))
10		nnacom 8557	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) = (𝑦 +o 𝐴))
11		suceq 6395	. . . . . . . . . . 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 2782	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 +o 𝑦) = suc (𝐴 +o 𝑦))
14		sseq2 3962	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 +o 𝑦) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (𝐴 +o 𝑦)))
15		suceq 6395	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐴 +o 𝑦) → suc 𝑥 = suc (𝐴 +o 𝑦))
16	15	eqeq2d 2748	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 +o 𝑦) → ((suc 𝐴 +o 𝑦) = suc 𝑥 ↔ (suc 𝐴 +o 𝑦) = suc (𝐴 +o 𝑦)))
17	14, 16	anbi12d 633	. . . . . . . . . 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 3578	. . . . . . . . 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 837	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ∃𝑥 ∈ ω (𝐴 ⊆ 𝑥 ∧ (suc 𝐴 +o 𝑦) = suc 𝑥))
20		eqeq1 2741	. . . . . . . . . 10 ⊢ ((suc 𝐴 +o 𝑦) = 𝐵 → ((suc 𝐴 +o 𝑦) = suc 𝑥 ↔ 𝐵 = suc 𝑥))
21	20	anbi2d 631	. . . . . . . . 9 ⊢ ((suc 𝐴 +o 𝑦) = 𝐵 → ((𝐴 ⊆ 𝑥 ∧ (suc 𝐴 +o 𝑦) = suc 𝑥) ↔ (𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥)))
22	21	rexbidv 3162	. . . . . . . 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 3139	. . . . . 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 7844	. . . . . . . 8 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
29	28	ad2antlr 728	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ⊆ 𝑥) → suc 𝑥 ∈ ω)
30		nnord 7828	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → Ord 𝐴)
31		nnord 7828	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → Ord 𝑥)
32		ordsucsssuc 7777	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ Ord 𝑥) → (𝐴 ⊆ 𝑥 ↔ suc 𝐴 ⊆ suc 𝑥))
33	30, 31, 32	syl2an 597	. . . . . . . 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 2825	. . . . . . 7 ⊢ (𝐵 = suc 𝑥 → (𝐵 ∈ ω ↔ suc 𝑥 ∈ ω))
37		sseq2 3962	. . . . . . 7 ⊢ (𝐵 = suc 𝑥 → (suc 𝐴 ⊆ 𝐵 ↔ suc 𝐴 ⊆ suc 𝑥))
38	36, 37	anbi12d 633	. . . . . 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 3139	. . 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 3913	. . 3 ⊢ (𝐵 ∈ (ω ∖ suc 𝐴) ↔ (𝐵 ∈ ω ∧ ¬ 𝐵 ∈ suc 𝐴))
44		nnord 7828	. . . . . 6 ⊢ (suc 𝐴 ∈ ω → Ord suc 𝐴)
45	1, 44	syl 17	. . . . 5 ⊢ (𝐴 ∈ ω → Ord suc 𝐴)
46		nnord 7828	. . . . 5 ⊢ (𝐵 ∈ ω → Ord 𝐵)
47		ordtri1 6360	. . . . 5 ⊢ ((Ord suc 𝐴 ∧ Ord 𝐵) → (suc 𝐴 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ suc 𝐴))
48	45, 46, 47	syl2an 597	. . . 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 3913	. . . . . 6 ⊢ (𝑥 ∈ (ω ∖ 𝐴) ↔ (𝑥 ∈ ω ∧ ¬ 𝑥 ∈ 𝐴))
52	51	anbi1i 625	. . . . 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 3081	. . 3 ⊢ (∃𝑥 ∈ (ω ∖ 𝐴)𝐵 = suc 𝑥 ↔ ∃𝑥 ∈ ω (¬ 𝑥 ∈ 𝐴 ∧ 𝐵 = suc 𝑥))
56		ordtri1 6360	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝑥) → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴))
57	30, 31, 56	syl2an 597	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴))
58	57	anbi1d 632	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 ⊆ 𝑥 ∧ 𝐵 = suc 𝑥) ↔ (¬ 𝑥 ∈ 𝐴 ∧ 𝐵 = suc 𝑥)))
59	58	rexbidva 3160	. . 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 1542   ∈ wcel 2114  ∃wrex 3062   ∖ cdif 3900   ⊆ wss 3903  Ord word 6326  suc csuc 6329  (class class class)co 7370  ωcom 7820   +_o coa 8406

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-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

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-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-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 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-oadd 8413

This theorem is referenced by:  brttrcl2  9637  ttrcltr  9639  rnttrcl  9645