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Theorem eldifsucnn 8702
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.
StepHypRef Expression
1 peano2 7912 . . . . . 6 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2 nnawordex 8675 . . . . . 6 ((suc 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴𝐵 ↔ ∃𝑦 ∈ ω (suc 𝐴 +o 𝑦) = 𝐵))
31, 2sylan 580 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴𝐵 ↔ ∃𝑦 ∈ ω (suc 𝐴 +o 𝑦) = 𝐵))
4 nnacl 8649 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) ∈ ω)
5 nnaword1 8667 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝑦))
6 nnasuc 8644 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +o suc 𝐴) = suc (𝑦 +o 𝐴))
76ancoms 458 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +o suc 𝐴) = suc (𝑦 +o 𝐴))
8 nnacom 8655 . . . . . . . . . . 11 ((suc 𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 +o 𝑦) = (𝑦 +o suc 𝐴))
91, 8sylan 580 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 +o 𝑦) = (𝑦 +o suc 𝐴))
10 nnacom 8655 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) = (𝑦 +o 𝐴))
11 suceq 6450 . . . . . . . . . . 11 ((𝐴 +o 𝑦) = (𝑦 +o 𝐴) → suc (𝐴 +o 𝑦) = suc (𝑦 +o 𝐴))
1210, 11syl 17 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝐴 +o 𝑦) = suc (𝑦 +o 𝐴))
137, 9, 123eqtr4d 2787 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 +o 𝑦) = suc (𝐴 +o 𝑦))
14 sseq2 4010 . . . . . . . . . . 11 (𝑥 = (𝐴 +o 𝑦) → (𝐴𝑥𝐴 ⊆ (𝐴 +o 𝑦)))
15 suceq 6450 . . . . . . . . . . . 12 (𝑥 = (𝐴 +o 𝑦) → suc 𝑥 = suc (𝐴 +o 𝑦))
1615eqeq2d 2748 . . . . . . . . . . 11 (𝑥 = (𝐴 +o 𝑦) → ((suc 𝐴 +o 𝑦) = suc 𝑥 ↔ (suc 𝐴 +o 𝑦) = suc (𝐴 +o 𝑦)))
1714, 16anbi12d 632 . . . . . . . . . 10 (𝑥 = (𝐴 +o 𝑦) → ((𝐴𝑥 ∧ (suc 𝐴 +o 𝑦) = suc 𝑥) ↔ (𝐴 ⊆ (𝐴 +o 𝑦) ∧ (suc 𝐴 +o 𝑦) = suc (𝐴 +o 𝑦))))
1817rspcev 3622 . . . . . . . . 9 (((𝐴 +o 𝑦) ∈ ω ∧ (𝐴 ⊆ (𝐴 +o 𝑦) ∧ (suc 𝐴 +o 𝑦) = suc (𝐴 +o 𝑦))) → ∃𝑥 ∈ ω (𝐴𝑥 ∧ (suc 𝐴 +o 𝑦) = suc 𝑥))
194, 5, 13, 18syl12anc 837 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ∃𝑥 ∈ ω (𝐴𝑥 ∧ (suc 𝐴 +o 𝑦) = suc 𝑥))
20 eqeq1 2741 . . . . . . . . . 10 ((suc 𝐴 +o 𝑦) = 𝐵 → ((suc 𝐴 +o 𝑦) = suc 𝑥𝐵 = suc 𝑥))
2120anbi2d 630 . . . . . . . . 9 ((suc 𝐴 +o 𝑦) = 𝐵 → ((𝐴𝑥 ∧ (suc 𝐴 +o 𝑦) = suc 𝑥) ↔ (𝐴𝑥𝐵 = suc 𝑥)))
2221rexbidv 3179 . . . . . . . 8 ((suc 𝐴 +o 𝑦) = 𝐵 → (∃𝑥 ∈ ω (𝐴𝑥 ∧ (suc 𝐴 +o 𝑦) = suc 𝑥) ↔ ∃𝑥 ∈ ω (𝐴𝑥𝐵 = suc 𝑥)))
2319, 22syl5ibcom 245 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((suc 𝐴 +o 𝑦) = 𝐵 → ∃𝑥 ∈ ω (𝐴𝑥𝐵 = suc 𝑥)))
2423rexlimdva 3155 . . . . . 6 (𝐴 ∈ ω → (∃𝑦 ∈ ω (suc 𝐴 +o 𝑦) = 𝐵 → ∃𝑥 ∈ ω (𝐴𝑥𝐵 = suc 𝑥)))
2524adantr 480 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑦 ∈ ω (suc 𝐴 +o 𝑦) = 𝐵 → ∃𝑥 ∈ ω (𝐴𝑥𝐵 = suc 𝑥)))
263, 25sylbid 240 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴𝐵 → ∃𝑥 ∈ ω (𝐴𝑥𝐵 = suc 𝑥)))
2726expimpd 453 . . 3 (𝐴 ∈ ω → ((𝐵 ∈ ω ∧ suc 𝐴𝐵) → ∃𝑥 ∈ ω (𝐴𝑥𝐵 = suc 𝑥)))
28 peano2 7912 . . . . . . . 8 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
2928ad2antlr 727 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴𝑥) → suc 𝑥 ∈ ω)
30 nnord 7895 . . . . . . . . 9 (𝐴 ∈ ω → Ord 𝐴)
31 nnord 7895 . . . . . . . . 9 (𝑥 ∈ ω → Ord 𝑥)
32 ordsucsssuc 7843 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝑥) → (𝐴𝑥 ↔ suc 𝐴 ⊆ suc 𝑥))
3330, 31, 32syl2an 596 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴𝑥 ↔ suc 𝐴 ⊆ suc 𝑥))
3433biimpa 476 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴𝑥) → suc 𝐴 ⊆ suc 𝑥)
3529, 34jca 511 . . . . . 6 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴𝑥) → (suc 𝑥 ∈ ω ∧ suc 𝐴 ⊆ suc 𝑥))
36 eleq1 2829 . . . . . . 7 (𝐵 = suc 𝑥 → (𝐵 ∈ ω ↔ suc 𝑥 ∈ ω))
37 sseq2 4010 . . . . . . 7 (𝐵 = suc 𝑥 → (suc 𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝑥))
3836, 37anbi12d 632 . . . . . 6 (𝐵 = suc 𝑥 → ((𝐵 ∈ ω ∧ suc 𝐴𝐵) ↔ (suc 𝑥 ∈ ω ∧ suc 𝐴 ⊆ suc 𝑥)))
3935, 38syl5ibrcom 247 . . . . 5 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴𝑥) → (𝐵 = suc 𝑥 → (𝐵 ∈ ω ∧ suc 𝐴𝐵)))
4039expimpd 453 . . . 4 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴𝑥𝐵 = suc 𝑥) → (𝐵 ∈ ω ∧ suc 𝐴𝐵)))
4140rexlimdva 3155 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ ω (𝐴𝑥𝐵 = suc 𝑥) → (𝐵 ∈ ω ∧ suc 𝐴𝐵)))
4227, 41impbid 212 . 2 (𝐴 ∈ ω → ((𝐵 ∈ ω ∧ suc 𝐴𝐵) ↔ ∃𝑥 ∈ ω (𝐴𝑥𝐵 = suc 𝑥)))
43 eldif 3961 . . 3 (𝐵 ∈ (ω ∖ suc 𝐴) ↔ (𝐵 ∈ ω ∧ ¬ 𝐵 ∈ suc 𝐴))
44 nnord 7895 . . . . . 6 (suc 𝐴 ∈ ω → Ord suc 𝐴)
451, 44syl 17 . . . . 5 (𝐴 ∈ ω → Ord suc 𝐴)
46 nnord 7895 . . . . 5 (𝐵 ∈ ω → Ord 𝐵)
47 ordtri1 6417 . . . . 5 ((Ord suc 𝐴 ∧ Ord 𝐵) → (suc 𝐴𝐵 ↔ ¬ 𝐵 ∈ suc 𝐴))
4845, 46, 47syl2an 596 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴𝐵 ↔ ¬ 𝐵 ∈ suc 𝐴))
4948pm5.32da 579 . . 3 (𝐴 ∈ ω → ((𝐵 ∈ ω ∧ suc 𝐴𝐵) ↔ (𝐵 ∈ ω ∧ ¬ 𝐵 ∈ suc 𝐴)))
5043, 49bitr4id 290 . 2 (𝐴 ∈ ω → (𝐵 ∈ (ω ∖ suc 𝐴) ↔ (𝐵 ∈ ω ∧ suc 𝐴𝐵)))
51 eldif 3961 . . . . . 6 (𝑥 ∈ (ω ∖ 𝐴) ↔ (𝑥 ∈ ω ∧ ¬ 𝑥𝐴))
5251anbi1i 624 . . . . 5 ((𝑥 ∈ (ω ∖ 𝐴) ∧ 𝐵 = suc 𝑥) ↔ ((𝑥 ∈ ω ∧ ¬ 𝑥𝐴) ∧ 𝐵 = suc 𝑥))
53 anass 468 . . . . 5 (((𝑥 ∈ ω ∧ ¬ 𝑥𝐴) ∧ 𝐵 = suc 𝑥) ↔ (𝑥 ∈ ω ∧ (¬ 𝑥𝐴𝐵 = suc 𝑥)))
5452, 53bitri 275 . . . 4 ((𝑥 ∈ (ω ∖ 𝐴) ∧ 𝐵 = suc 𝑥) ↔ (𝑥 ∈ ω ∧ (¬ 𝑥𝐴𝐵 = suc 𝑥)))
5554rexbii2 3090 . . 3 (∃𝑥 ∈ (ω ∖ 𝐴)𝐵 = suc 𝑥 ↔ ∃𝑥 ∈ ω (¬ 𝑥𝐴𝐵 = suc 𝑥))
56 ordtri1 6417 . . . . . 6 ((Ord 𝐴 ∧ Ord 𝑥) → (𝐴𝑥 ↔ ¬ 𝑥𝐴))
5730, 31, 56syl2an 596 . . . . 5 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴𝑥 ↔ ¬ 𝑥𝐴))
5857anbi1d 631 . . . 4 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴𝑥𝐵 = suc 𝑥) ↔ (¬ 𝑥𝐴𝐵 = suc 𝑥)))
5958rexbidva 3177 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ ω (𝐴𝑥𝐵 = suc 𝑥) ↔ ∃𝑥 ∈ ω (¬ 𝑥𝐴𝐵 = suc 𝑥)))
6055, 59bitr4id 290 . 2 (𝐴 ∈ ω → (∃𝑥 ∈ (ω ∖ 𝐴)𝐵 = suc 𝑥 ↔ ∃𝑥 ∈ ω (𝐴𝑥𝐵 = suc 𝑥)))
6142, 50, 603bitr4d 311 1 (𝐴 ∈ ω → (𝐵 ∈ (ω ∖ suc 𝐴) ↔ ∃𝑥 ∈ (ω ∖ 𝐴)𝐵 = suc 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  cdif 3948  wss 3951  Ord word 6383  suc csuc 6386  (class class class)co 7431  ωcom 7887   +o coa 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510
This theorem is referenced by:  brttrcl2  9754  ttrcltr  9756  rnttrcl  9762
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