Theorem onun2 4491

Description: The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.)

Assertion

Ref	Expression
onun2	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)

Proof of Theorem onun2

Step	Hyp	Ref	Expression
1		prssi 3752	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ⊆ On)
2		prexg 4213	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ∈ V)
3		ssonuni 4489	. . . 4 ⊢ ({𝐴, 𝐵} ∈ V → ({𝐴, 𝐵} ⊆ On → ∪ {𝐴, 𝐵} ∈ On))
4	2, 3	syl 14	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴, 𝐵} ⊆ On → ∪ {𝐴, 𝐵} ∈ On))
5		uniprg 3826	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
6	5	eleq1d 2246	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∪ {𝐴, 𝐵} ∈ On ↔ (𝐴 ∪ 𝐵) ∈ On))
7	4, 6	sylibd 149	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴, 𝐵} ⊆ On → (𝐴 ∪ 𝐵) ∈ On))
8	1, 7	mpd 13	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2148  Vcvv 2739   ∪ cun 3129   ⊆ wss 3131  {cpr 3595  ∪ cuni 3811  Oncon0 4365

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-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370

This theorem is referenced by:  onun2i  4492  rdgon  6389