Theorem rext 4214

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 3624	. . 3 ⊢ 𝑥 ∈ {𝑥}
2		vex 2740	. . . . 5 ⊢ 𝑥 ∈ V
3	2	snex 4184	. . . 4 ⊢ {𝑥} ∈ V
4		eleq2 2241	. . . . 5 ⊢ (𝑧 = {𝑥} → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ {𝑥}))
5		eleq2 2241	. . . . 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 2831	. . 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 3609	. . 3 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
10		equcomi 1704	. . 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 1351   = wceq 1353   ∈ wcel 2148  {csn 3592

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598

This theorem is referenced by: (None)