Theorem onunsnss 7218

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 4686	. . . . 5 ⊢ ¬ 𝐵 ∈ 𝐵
2		elsni 3726	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
3	2	adantl 277	. . . . . . 7 ⊢ ((((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝑥 = 𝐵)
4		simplr 533	. . . . . . 7 ⊢ ((((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝑥 ∈ 𝐵)
5	3, 4	eqeltrrd 2316	. . . . . 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 674	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ {𝐵})
8		snidg 3737	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
9		elun2 3397	. . . . . . . . 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 4532	. . . . . . . 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 432	. . . . . 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 3370	. . . . 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 1390	. . 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 3254	1 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 720   = wceq 1402   ∈ wcel 2209   ∪ cun 3218   ⊆ wss 3220  {csn 3708  Oncon0 4506

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-sn 3714  df-uni 3934  df-tr 4228  df-iord 4509  df-on 4511

This theorem is referenced by: (None)