Theorem onunsnss 6973

Description: Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)

Assertion

Ref	Expression
onunsnss	⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵 ⊆ 𝐴)

Proof of Theorem onunsnss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elirr 4573	. . . . 5 ⊢ ¬ 𝐵 ∈ 𝐵
2		elsni 3636	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
3	2	adantl 277	. . . . . . 7 ⊢ ((((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝑥 = 𝐵)
4		simplr 528	. . . . . . 7 ⊢ ((((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝑥 ∈ 𝐵)
5	3, 4	eqeltrrd 2271	. . . . . 6 ⊢ ((((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝐵 ∈ 𝐵)
6	5	ex 115	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ {𝐵} → 𝐵 ∈ 𝐵))
7	1, 6	mtoi 665	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ {𝐵})
8		snidg 3647	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
9		elun2 3327	. . . . . . . . 9 ⊢ (𝐵 ∈ {𝐵} → 𝐵 ∈ (𝐴 ∪ {𝐵}))
10	8, 9	syl 14	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ (𝐴 ∪ {𝐵}))
11	10	adantr 276	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
12		ontr1 4420	. . . . . . . 8 ⊢ ((𝐴 ∪ {𝐵}) ∈ On → ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵})))
13	12	adantl 277	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵})))
14	11, 13	mpan2d 428	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐴 ∪ {𝐵})))
15	14	imp 124	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐴 ∪ {𝐵}))
16		elun 3300	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵}))
17	15, 16	sylib 122	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵}))
18	7, 17	ecased 1360	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
19	18	ex 115	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))
20	19	ssrdv 3185	1 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2164   ∪ cun 3151   ⊆ wss 3153  {csn 3618  Oncon0 4394

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-in1 615  ax-in2 616  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-ext 2175  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399

This theorem is referenced by: (None)