Theorem sucunielr 4511

Description: Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4532. (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 2750	. . . 4 ⊢ (suc 𝐴 ∈ 𝐵 → suc 𝐴 ∈ V)
2		sucexb 4498	. . . 4 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylibr 134	. . 3 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ V)
4		sucidg 4418	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5	3, 4	syl 14	. 2 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐴)
6		elunii 3816	. 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 2148  Vcvv 2739  ∪ cuni 3811  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-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-suc 4373

This theorem is referenced by:  nnsucuniel  6499