Theorem onunsnss 7179

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 4665	. . . . 5 ⊢ ¬ 𝐵 ∈ 𝐵
2		elsni 3709	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
3	2	adantl 277	. . . . . . 7 ⊢ ((((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝑥 = 𝐵)
4		simplr 529	. . . . . . 7 ⊢ ((((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝑥 ∈ 𝐵)
5	3, 4	eqeltrrd 2312	. . . . . 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 670	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ {𝐵})
8		snidg 3720	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵})
9		elun2 3389	. . . . . . . . 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 4512	. . . . . . . 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 3362	. . . . 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 1386	. . 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 3246	1 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398   ∈ wcel 2205   ∪ cun 3211   ⊆ wss 3213  {csn 3691  Oncon0 4486

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-sn 3697  df-uni 3917  df-tr 4211  df-iord 4489  df-on 4491

This theorem is referenced by: (None)