Theorem axprALT 5288

Description: Alternate proof of axpr 5294. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
axprALT	⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)

Distinct variable groups:   𝑥,𝑧,𝑤   𝑦,𝑧,𝑤

Proof of Theorem axprALT

Step	Hyp	Ref	Expression
1		zfpair 5287	. . 3 ⊢ {𝑥, 𝑦} ∈ V
2	1	isseti 3455	. 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦}
3		dfcleq 2792	. . 3 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}))
4		vex 3444	. . . . . . 7 ⊢ 𝑤 ∈ V
5	4	elpr 4548	. . . . . 6 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
6	5	bibi2i 341	. . . . 5 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
7		biimpr 223	. . . . 5 ⊢ ((𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) → ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))
8	6, 7	sylbi 220	. . . 4 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) → ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))
9	8	alimi 1813	. . 3 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) → ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))
10	3, 9	sylbi 220	. 2 ⊢ (𝑧 = {𝑥, 𝑦} → ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))
11	2, 10	eximii 1838	1 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)

Syntax hints:   → wi 4   ↔ wb 209   ∨ wo 844  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cpr 4527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-pw 4499  df-sn 4526  df-pr 4528

This theorem is referenced by: (None)