Theorem onun2 4414

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 3686	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ⊆ On)
2		prexg 4141	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ∈ V)
3		ssonuni 4412	. . . 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 3759	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
6	5	eleq1d 2209	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∪ {𝐴, 𝐵} ∈ On ↔ (𝐴 ∪ 𝐵) ∈ On))
7	4, 6	sylibd 148	. 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 103   ∈ wcel 1481  Vcvv 2689   ∪ cun 3074   ⊆ wss 3076  {cpr 3533  ∪ cuni 3744  Oncon0 4293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-uni 3745  df-tr 4035  df-iord 4296  df-on 4298

This theorem is referenced by:  onun2i  4415  rdgon  6291