Theorem elALT2 5340

Description: Alternate proof of el 5419 using ax-9 2153 and ax-pow 5336 instead of ax-pr 5404. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elALT2	⊢ ∃𝑦 𝑥 ∈ 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem elALT2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfpow 5337	. 2 ⊢ ∃𝑦∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)
2		ax9 2157	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥))
3	2	alrimiv 1957	. . . 4 ⊢ (𝑧 = 𝑥 → ∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥))
4		ax8 2149	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 → 𝑥 ∈ 𝑦))
5	3, 4	embantd 60	. . 3 ⊢ (𝑧 = 𝑥 → ((∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦))
6	5	spimvw 2016	. 2 ⊢ (∀𝑧(∀𝑦(𝑦 ∈ 𝑧 → 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦)
7	1, 6	eximii 1867	1 ⊢ ∃𝑦 𝑥 ∈ 𝑦

Syntax hints:   → wi 4  ∀wal 1568  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-pow 5336

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is referenced by:  dtruALT2  5341  dvdemo2  5345