Theorem rext 4258

Description: A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)

Assertion

Ref	Expression
rext	⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦)

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem rext

Step	Hyp	Ref	Expression
1		vsnid 3664	. . 3 ⊢ 𝑥 ∈ {𝑥}
2		vex 2774	. . . . 5 ⊢ 𝑥 ∈ V
3	2	snex 4228	. . . 4 ⊢ {𝑥} ∈ V
4		eleq2 2268	. . . . 5 ⊢ (𝑧 = {𝑥} → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ {𝑥}))
5		eleq2 2268	. . . . 5 ⊢ (𝑧 = {𝑥} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑥}))
6	4, 5	imbi12d 234	. . . 4 ⊢ (𝑧 = {𝑥} → ((𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) ↔ (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥})))
7	3, 6	spcv 2866	. . 3 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥}))
8	1, 7	mpi 15	. 2 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑦 ∈ {𝑥})
9		velsn 3649	. . 3 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
10		equcomi 1726	. . 3 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
11	9, 10	sylbi 121	. 2 ⊢ (𝑦 ∈ {𝑥} → 𝑥 = 𝑦)
12	8, 11	syl 14	1 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦)

Syntax hints:   → wi 4  ∀wal 1370   = wceq 1372   ∈ wcel 2175  {csn 3632

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-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638

This theorem is referenced by: (None)