Theorem onsucelsucr 4606

Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4628. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6658. (Contributed by Jim Kingdon, 17-Jul-2019.)

Assertion

Ref	Expression
onsucelsucr	⊢ (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ 𝐵))

Proof of Theorem onsucelsucr

Step	Hyp	Ref	Expression
1		elex 2814	. . . 4 ⊢ (suc 𝐴 ∈ suc 𝐵 → suc 𝐴 ∈ V)
2		sucexb 4595	. . . 4 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylibr 134	. . 3 ⊢ (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ V)
4		onelss 4484	. . . . . . 7 ⊢ (𝐵 ∈ On → (suc 𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
5		eqimss 3281	. . . . . . . 8 ⊢ (suc 𝐴 = 𝐵 → suc 𝐴 ⊆ 𝐵)
6	5	a1i 9	. . . . . . 7 ⊢ (𝐵 ∈ On → (suc 𝐴 = 𝐵 → suc 𝐴 ⊆ 𝐵))
7	4, 6	jaod 724	. . . . . 6 ⊢ (𝐵 ∈ On → ((suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵) → suc 𝐴 ⊆ 𝐵))
8	7	adantl 277	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ On) → ((suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵) → suc 𝐴 ⊆ 𝐵))
9		elsucg 4501	. . . . . . 7 ⊢ (suc 𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵)))
10	2, 9	sylbi 121	. . . . . 6 ⊢ (𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵)))
11	10	adantr 276	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴 ∈ 𝐵 ∨ suc 𝐴 = 𝐵)))
12		eloni 4472	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
13		ordelsuc 4603	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵))
14	12, 13	sylan2 286	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ suc 𝐴 ⊆ 𝐵))
15	8, 11, 14	3imtr4d 203	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ 𝐵))
16	15	impancom 260	. . 3 ⊢ ((𝐴 ∈ V ∧ suc 𝐴 ∈ suc 𝐵) → (𝐵 ∈ On → 𝐴 ∈ 𝐵))
17	3, 16	mpancom 422	. 2 ⊢ (suc 𝐴 ∈ suc 𝐵 → (𝐵 ∈ On → 𝐴 ∈ 𝐵))
18	17	com12 30	1 ⊢ (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ⊆ wss 3200  Ord word 4459  Oncon0 4460  suc csuc 4462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465  df-suc 4468

This theorem is referenced by:  nnsucelsuc  6658