Theorem sucunielr 4557

Description: Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4578. (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 2782	. . . 4 ⊢ (suc 𝐴 ∈ 𝐵 → suc 𝐴 ∈ V)
2		sucexb 4544	. . . 4 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylibr 134	. . 3 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ V)
4		sucidg 4462	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5	3, 4	syl 14	. 2 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐴)
6		elunii 3854	. 2 ⊢ ((𝐴 ∈ suc 𝐴 ∧ suc 𝐴 ∈ 𝐵) → 𝐴 ∈ ∪ 𝐵)
7	5, 6	mpancom 422	1 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ ∪ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2175  Vcvv 2771  ∪ cuni 3849  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-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-suc 4417

This theorem is referenced by:  nnsucuniel  6580