Theorem sucunielr 4546

Description: Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4567. (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 2774	. . . 4 ⊢ (suc 𝐴 ∈ 𝐵 → suc 𝐴 ∈ V)
2		sucexb 4533	. . . 4 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
3	1, 2	sylibr 134	. . 3 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ V)
4		sucidg 4451	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
5	3, 4	syl 14	. 2 ⊢ (suc 𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐴)
6		elunii 3844	. 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 2167  Vcvv 2763  ∪ cuni 3839  suc csuc 4400

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-suc 4406

This theorem is referenced by:  nnsucuniel  6553