Theorem sucunielr 4421

Description: Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4441. (Contributed by Jim Kingdon, 2-Aug-2019.)

Assertion

Ref	Expression
sucunielr	⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵)

Proof of Theorem sucunielr

Step	Hyp	Ref	Expression
1		elex 2692	. . . 4 ⊢ (suc 𝐴 ∈ 𝐵 → suc 𝐴 ∈ V)
2		sucexb 4408	. . . 4 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylibr 133	. . 3 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ V)
4		sucidg 4333	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5	3, 4	syl 14	. 2 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐴)
6		elunii 3736	. 2 ⊢ ((𝐴 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝐵) → 𝐴 ∈ ∪ 𝐵)
7	5, 6	mpancom 418	1 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 1480  Vcvv 2681  ∪ cuni 3731  suc csuc 4282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-suc 4288

This theorem is referenced by:  nnsucuniel  6384