Theorem succlg 43327

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 2824	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ ∅))
2		noel 4318	. . . . . 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 4111	. . . . . . . . 9 ⊢ (𝐶 ∈ (On ∖ 1o) → 𝐶 ∈ On)
8	7	ad2antll 729	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → 𝐶 ∈ On)
9		omex 9662	. . . . . . . . . 10 ⊢ ω ∈ V
10		limom 7882	. . . . . . . . . 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 8518	. . . . . . . . . 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 43241	. . . . . . . 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 6369	. . . . . . . 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 7849	. . . . . 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 1540   ∈ wcel 2109  Vcvv 3464   ∖ cdif 3928  ∅c0 4313  Oncon0 6357  Lim wlim 6358  suc csuc 6359  (class class class)co 7410  ωcom 7866  1_oc1o 8478   ·_o comu 8483

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734  ax-inf2 9660

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-omul 8490

This theorem is referenced by: (None)