Theorem onsucelsucr 4555

Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4577. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6576. (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 2782	. . . 4 ⊢ (suc 𝐴 ∈ suc 𝐵 → suc 𝐴 ∈ V)
2		sucexb 4544	. . . 4 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylibr 134	. . 3 ⊢ (suc 𝐴 ∈ suc 𝐵 → 𝐴 ∈ V)
4		onelss 4433	. . . . . . 7 ⊢ (𝐵 ∈ On → (suc 𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵))
5		eqimss 3246	. . . . . . . 8 ⊢ (suc 𝐴 = 𝐵 → suc 𝐴 ⊆ 𝐵)
6	5	a1i 9	. . . . . . 7 ⊢ (𝐵 ∈ On → (suc 𝐴 = 𝐵 → suc 𝐴 ⊆ 𝐵))
7	4, 6	jaod 718	. . . . . 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 4450	. . . . . . 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 4421	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
13		ordelsuc 4552	. . . . . 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 709   = wceq 1372   ∈ wcel 2175  Vcvv 2771   ⊆ wss 3165  Ord word 4408  Oncon0 4409  suc csuc 4411

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-uni 3850  df-tr 4142  df-iord 4412  df-on 4414  df-suc 4417

This theorem is referenced by:  nnsucelsuc  6576