Theorem snon0 6356

Description: An ordinal which is a singleton is {∅}. (Contributed by Jim Kingdon, 19-Oct-2021.)

Assertion

Ref	Expression
snon0	⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅)

Proof of Theorem snon0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elirr 4266	. . 3 ⊢ ¬ 𝐴 ∈ 𝐴
2		snidg 3400	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
3	2	adantr 261	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 ∈ {𝐴})
4		ontr1 4126	. . . . . . 7 ⊢ ({𝐴} ∈ On → ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ {𝐴}))
5	4	adantl 262	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ {𝐴}))
6	3, 5	mpan2d 404	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐴}))
7		elsni 3393	. . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
8	6, 7	syl6 29	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → (𝑥 ∈ 𝐴 → 𝑥 = 𝐴))
9		eleq1 2100	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
10	9	biimpcd 148	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝐴 → 𝐴 ∈ 𝐴))
11	8, 10	sylcom 25	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → (𝑥 ∈ 𝐴 → 𝐴 ∈ 𝐴))
12	1, 11	mtoi 590	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → ¬ 𝑥 ∈ 𝐴)
13	12	eq0rdv 3261	1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∅c0 3224  {csn 3375  Oncon0 4100

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105

This theorem is referenced by: (None)