Theorem succlg 44331

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 2850	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ ∅))
2		noel 4284	. . . . . 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 488	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → 𝐴 ∈ 𝐵)
7		eldifi 4078	. . . . . . . . 9 ⊢ (𝐶 ∈ (On ∖ 1o) → 𝐶 ∈ On)
8	7	ad2antll 742	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → 𝐶 ∈ On)
9		omex 9645	. . . . . . . . . 10 ⊢ ω ∈ V
10		limom 7894	. . . . . . . . . 10 ⊢ Lim ω
11	9, 10	pm3.2i 476	. . . . . . . . 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 8510	. . . . . . . . . 10 ⊢ (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
14	13	simprbi 503	. . . . . . . . 9 ⊢ (𝐶 ∈ (On ∖ 1o) → ∅ ∈ 𝐶)
15	14	ad2antll 742	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → ∅ ∈ 𝐶)
16		omlimcl2 44245	. . . . . . . 8 ⊢ (((𝐶 ∈ On ∧ (ω ∈ V ∧ Lim ω)) ∧ ∅ ∈ 𝐶) → Lim (ω ·o 𝐶))
17	8, 12, 15, 16	syl21anc 851	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → Lim (ω ·o 𝐶))
18		limeq 6375	. . . . . . . 8 ⊢ (𝐵 = (ω ·o 𝐶) → (Lim 𝐵 ↔ Lim (ω ·o 𝐶)))
19	18	ad2antrl 741	. . . . . . 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 7861	. . . . . 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 418	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o)) → suc 𝐴 ∈ 𝐵))
25	5, 24	jaod 873	. 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐵 = ∅ ∨ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o))) → suc 𝐴 ∈ 𝐵))
26	25	imp 412	1 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = ∅ ∨ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o)))) → suc 𝐴 ∈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∖ cdif 3896  ∅c0 4279  Oncon0 6363  Lim wlim 6364  suc csuc 6365  (class class class)co 7421  ωcom 7878  1_oc1o 8470   ·_o comu 8475

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7752  ax-inf2 9643

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3900  df-un 3902  df-in 3904  df-ss 3915  df-pss 4035  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-2nd 8003  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-oadd 8481  df-omul 8482

This theorem is used by: (None)