Theorem snon0 6913

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 4525	. . 3 ⊢ ¬ 𝐴 ∈ 𝐴
2		snidg 3612	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
3	2	adantr 274	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 ∈ {𝐴})
4		ontr1 4374	. . . . . . 7 ⊢ ({𝐴} ∈ On → ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ {𝐴}))
5	4	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ {𝐴}))
6	3, 5	mpan2d 426	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐴}))
7		elsni 3601	. . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
8	6, 7	syl6 33	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → (𝑥 ∈ 𝐴 → 𝑥 = 𝐴))
9		eleq1 2233	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
10	9	biimpcd 158	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝐴 → 𝐴 ∈ 𝐴))
11	8, 10	sylcom 28	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → (𝑥 ∈ 𝐴 → 𝐴 ∈ 𝐴))
12	1, 11	mtoi 659	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → ¬ 𝑥 ∈ 𝐴)
13	12	eq0rdv 3459	1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  ∅c0 3414  {csn 3583  Oncon0 4348

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-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353

This theorem is referenced by: (None)