Theorem snon0 7001

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 4577	. . 3 ⊢ ¬ 𝐴 ∈ 𝐴
2		snidg 3651	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
3	2	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 ∈ {𝐴})
4		ontr1 4424	. . . . . . 7 ⊢ ({𝐴} ∈ On → ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ {𝐴}))
5	4	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ {𝐴}) → 𝑥 ∈ {𝐴}))
6	3, 5	mpan2d 428	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐴}))
7		elsni 3640	. . . . 5 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
8	6, 7	syl6 33	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → (𝑥 ∈ 𝐴 → 𝑥 = 𝐴))
9		eleq1 2259	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
10	9	biimpcd 159	. . . 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 665	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → ¬ 𝑥 ∈ 𝐴)
13	12	eq0rdv 3495	1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∅c0 3450  {csn 3622  Oncon0 4398

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451  df-sn 3628  df-uni 3840  df-tr 4132  df-iord 4401  df-on 4403

This theorem is referenced by: (None)