Theorem axprALT 5378

Description: Alternate proof of axpr 5383. (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 5377	. . 3 ⊢ {𝑥, 𝑦} ∈ V
2	1	isseti 3471	. 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦}
3		dfcleq 2754	. . 3 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}))
4		vex 3457	. . . . . . 7 ⊢ 𝑤 ∈ V
5	4	elpr 4606	. . . . . 6 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
6	5	bibi2i 339	. . . . 5 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)))
7		biimpr 222	. . . . 5 ⊢ ((𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) → ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))
8	6, 7	sylbi 219	. . . 4 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) → ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))
9	8	alimi 1830	. . 3 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) → ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))
10	3, 9	sylbi 219	. 2 ⊢ (𝑧 = {𝑥, 𝑦} → ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))
11	2, 10	eximii 1856	1 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)

Syntax hints:   → wi 4   ↔ wb 208   ∨ wo 858  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cpr 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-pw 4556  df-sn 4582  df-pr 4584

This theorem is referenced by: (None)