Theorem onun2 4401

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 3673	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ⊆ On)
2		prexg 4128	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝐴, 𝐵} ∈ V)
3		ssonuni 4399	. . . 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 3746	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
6	5	eleq1d 2206	. . 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 1480  Vcvv 2681   ∪ cun 3064   ⊆ wss 3066  {cpr 3523  ∪ cuni 3731  Oncon0 4280

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285

This theorem is referenced by:  onun2i  4402  rdgon  6276