Theorem succlg 43512

Description: Closure law for ordinal successor. (Contributed by RP, 8-Jan-2025.)

Assertion

Ref	Expression
succlg	⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = ∅ ∨ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o)))) → suc 𝐴 ∈ 𝐵)

Proof of Theorem succlg

Step	Hyp	Ref	Expression
1		eleq2 2823	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ ∅))
2		noel 4288	. . . . . 6 ⊢ ¬ 𝐴 ∈ ∅
3	2	pm2.21i 119	. . . . 5 ⊢ (𝐴 ∈ ∅ → suc 𝐴 ∈ 𝐵)
4	1, 3	biimtrdi 253	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ∈ 𝐵 → suc 𝐴 ∈ 𝐵))
5	4	com12 32	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = ∅ → suc 𝐴 ∈ 𝐵))
6		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → 𝐴 ∈ 𝐵)
7		eldifi 4081	. . . . . . . . 9 ⊢ (𝐶 ∈ (On ∖ 1o) → 𝐶 ∈ On)
8	7	ad2antll 729	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → 𝐶 ∈ On)
9		omex 9550	. . . . . . . . . 10 ⊢ ω ∈ V
10		limom 7822	. . . . . . . . . 10 ⊢ Lim ω
11	9, 10	pm3.2i 470	. . . . . . . . 9 ⊢ (ω ∈ V ∧ Lim ω)
12	11	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → (ω ∈ V ∧ Lim ω))
13		ondif1 8426	. . . . . . . . . 10 ⊢ (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
14	13	simprbi 496	. . . . . . . . 9 ⊢ (𝐶 ∈ (On ∖ 1o) → ∅ ∈ 𝐶)
15	14	ad2antll 729	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → ∅ ∈ 𝐶)
16		omlimcl2 43426	. . . . . . . 8 ⊢ (((𝐶 ∈ On ∧ (ω ∈ V ∧ Lim ω)) ∧ ∅ ∈ 𝐶) → Lim (ω ·o 𝐶))
17	8, 12, 15, 16	syl21anc 837	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → Lim (ω ·o 𝐶))
18		limeq 6327	. . . . . . . 8 ⊢ (𝐵 = (ω ·o 𝐶) → (Lim 𝐵 ↔ Lim (ω ·o 𝐶)))
19	18	ad2antrl 728	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → (Lim 𝐵 ↔ Lim (ω ·o 𝐶)))
20	17, 19	mpbird 257	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → Lim 𝐵)
21		limsuc 7789	. . . . . 6 ⊢ (Lim 𝐵 → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ 𝐵))
22	20, 21	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ 𝐵))
23	6, 22	mpbid 232	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → suc 𝐴 ∈ 𝐵)
24	23	ex 412	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o)) → suc 𝐴 ∈ 𝐵))
25	5, 24	jaod 859	. 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐵 = ∅ ∨ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → suc 𝐴 ∈ 𝐵))
26	25	imp 406	1 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = ∅ ∨ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o)))) → suc 𝐴 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ∖ cdif 3896  ∅c0 4283  Oncon0 6315  Lim wlim 6316  suc csuc 6317  (class class class)co 7356  ωcom 7806  1_oc1o 8388   ·_o comu 8393

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678  ax-inf2 9548

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-omul 8400

This theorem is referenced by: (None)