Theorem onsucelsucr 4509

Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4531. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6495. (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 2750	. . . 4 ⊢ (suc 𝐴 ∈ suc 𝐵 → suc 𝐴 ∈ V)
2		sucexb 4498	. . . 4 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylibr 134	. . 3 ⊢ (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ V)
4		onelss 4389	. . . . . . 7 ⊢ (𝐵 ∈ On → (suc 𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
5		eqimss 3211	. . . . . . . 8 ⊢ (suc 𝐴 = 𝐵 → suc 𝐴 ⊆ 𝐵)
6	5	a1i 9	. . . . . . 7 ⊢ (𝐵 ∈ On → (suc 𝐴 = 𝐵 → suc 𝐴 ⊆ 𝐵))
7	4, 6	jaod 717	. . . . . 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 4406	. . . . . . 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 4377	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
13		ordelsuc 4506	. . . . . 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 708   = wceq 1353   ∈ wcel 2148  Vcvv 2739   ⊆ wss 3131  Ord word 4364  Oncon0 4365  suc csuc 4367

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370  df-suc 4373

This theorem is referenced by:  nnsucelsuc  6495