Theorem succlg 42011

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 4329	. . . . . 6 ⊢ ¬ 𝐴 ∈ ∅
3	2	pm2.21i 119	. . . . 5 ⊢ (𝐴 ∈ ∅ → suc 𝐴 ∈ 𝐵)
4	1, 3	syl6bi 253	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ∈ 𝐵 → suc 𝐴 ∈ 𝐵))
5	4	com12 32	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = ∅ → suc 𝐴 ∈ 𝐵))
6		simpl 484	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → 𝐴 ∈ 𝐵)
7		eldifi 4125	. . . . . . . . 9 ⊢ (𝐶 ∈ (On ∖ 1o) → 𝐶 ∈ On)
8	7	ad2antll 728	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → 𝐶 ∈ On)
9		omex 9634	. . . . . . . . . 10 ⊢ ω ∈ V
10		limom 7866	. . . . . . . . . 10 ⊢ Lim ω
11	9, 10	pm3.2i 472	. . . . . . . . 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 8496	. . . . . . . . . 10 ⊢ (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
14	13	simprbi 498	. . . . . . . . 9 ⊢ (𝐶 ∈ (On ∖ 1o) → ∅ ∈ 𝐶)
15	14	ad2antll 728	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → ∅ ∈ 𝐶)
16		omlimcl2 41924	. . . . . . . 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 6373	. . . . . . . 8 ⊢ (𝐵 = (ω ·o 𝐶) → (Lim 𝐵 ↔ Lim (ω ·o 𝐶)))
19	18	ad2antrl 727	. . . . . . 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 7833	. . . . . 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 231	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → suc 𝐴 ∈ 𝐵)
24	23	ex 414	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o)) → suc 𝐴 ∈ 𝐵))
25	5, 24	jaod 858	. 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐵 = ∅ ∨ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → suc 𝐴 ∈ 𝐵))
26	25	imp 408	1 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = ∅ ∨ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o)))) → suc 𝐴 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∖ cdif 3944  ∅c0 4321  Oncon0 6361  Lim wlim 6362  suc csuc 6363  (class class class)co 7404  ωcom 7850  1_oc1o 8454   ·_o comu 8459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720  ax-inf2 9632

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-omul 8466

This theorem is referenced by: (None)