Theorem succlg 44075

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 2852	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ ∅))
2		noel 4291	. . . . . 6 ⊢ ¬ 𝐴 ∈ ∅
3	2	pm2.21i 120	. . . . 5 ⊢ (𝐴 ∈ ∅ → suc 𝐴 ∈ 𝐵)
4	1, 3	biimtrdi 256	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ∈ 𝐵 → suc 𝐴 ∈ 𝐵))
5	4	com12 33	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = ∅ → suc 𝐴 ∈ 𝐵))
6		simpl 487	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → 𝐴 ∈ 𝐵)
7		eldifi 4085	. . . . . . . . 9 ⊢ (𝐶 ∈ (On ∖ 1o) → 𝐶 ∈ On)
8	7	ad2antll 741	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → 𝐶 ∈ On)
9		omex 9608	. . . . . . . . . 10 ⊢ ω ∈ V
10		limom 7874	. . . . . . . . . 10 ⊢ Lim ω
11	9, 10	pm3.2i 475	. . . . . . . . 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 8482	. . . . . . . . . 10 ⊢ (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
14	13	simprbi 502	. . . . . . . . 9 ⊢ (𝐶 ∈ (On ∖ 1o) → ∅ ∈ 𝐶)
15	14	ad2antll 741	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → ∅ ∈ 𝐶)
16		omlimcl2 43989	. . . . . . . 8 ⊢ (((𝐶 ∈ On ∧ (ω ∈ V ∧ Lim ω)) ∧ ∅ ∈ 𝐶) → Lim (ω ·o 𝐶))
17	8, 12, 15, 16	syl21anc 850	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → Lim (ω ·o 𝐶))
18		limeq 6372	. . . . . . . 8 ⊢ (𝐵 = (ω ·o 𝐶) → (Lim 𝐵 ↔ Lim (ω ·o 𝐶)))
19	18	ad2antrl 740	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → (Lim 𝐵 ↔ Lim (ω ·o 𝐶)))
20	17, 19	mpbird 260	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → Lim 𝐵)
21		limsuc 7841	. . . . . 6 ⊢ (Lim 𝐵 → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ 𝐵))
22	20, 21	syl 18	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ∈ 𝐵))
23	6, 22	mpbid 235	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → suc 𝐴 ∈ 𝐵)
24	23	ex 417	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o)) → suc 𝐴 ∈ 𝐵))
25	5, 24	jaod 872	. 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐵 = ∅ ∨ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → suc 𝐴 ∈ 𝐵))
26	25	imp 411	1 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = ∅ ∨ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o)))) → suc 𝐴 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3902  ∅c0 4286  Oncon0 6360  Lim wlim 6361  suc csuc 6362  (class class class)co 7410  ωcom 7858  1_oc1o 8442   ·_o comu 8447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  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 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-omul 8454

This theorem is referenced by: (None)