Theorem sucunielr 4527

Description: Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4548. (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 2763	. . . 4 ⊢ (suc 𝐴 ∈ 𝐵 → suc 𝐴 ∈ V)
2		sucexb 4514	. . . 4 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylibr 134	. . 3 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ V)
4		sucidg 4434	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5	3, 4	syl 14	. 2 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐴)
6		elunii 3829	. 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 2160  Vcvv 2752  ∪ cuni 3824  suc csuc 4383

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-suc 4389

This theorem is referenced by:  nnsucuniel  6521