Theorem sn-exelALT 40396

Description: Alternate proof of exel 5365, avoiding ax-pr 5361 but requiring ax-5 1911, ax-9 2114, and ax-pow 5297. This is similar to how elALT2 5301 uses ax-pow 5297 instead of ax-pr 5361 compared to el 5370. (Contributed by SN, 18-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sn-exelALT	⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem sn-exelALT

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-pow 5297	. 2 ⊢ ∃𝑦∀𝑥(∀𝑤(𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑧) → 𝑥 ∈ 𝑦)
2		ax6ev 1971	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑧
3		ax9v1 2116	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑧))
4	3	alrimiv 1928	. . . 4 ⊢ (𝑥 = 𝑧 → ∀𝑤(𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑧))
5	2, 4	eximii 1837	. . 3 ⊢ ∃𝑥∀𝑤(𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑧)
6		exim 1834	. . 3 ⊢ (∀𝑥(∀𝑤(𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑧) → 𝑥 ∈ 𝑦) → (∃𝑥∀𝑤(𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑧) → ∃𝑥 𝑥 ∈ 𝑦))
7	5, 6	mpi 20	. 2 ⊢ (∀𝑥(∀𝑤(𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑧) → 𝑥 ∈ 𝑦) → ∃𝑥 𝑥 ∈ 𝑦)
8	1, 7	eximii 1837	1 ⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-9 2114  ax-pow 5297

This theorem depends on definitions:  df-bi 206  df-ex 1780

This theorem is referenced by: (None)