Theorem axprlem5OLD 5363

Description: Obsolete version of axprlem4 5358 as of 18-Sep-2025. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axprlem5OLD	⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))

Distinct variable groups:   𝑦,𝑠   𝑤,𝑠   𝑛,𝑠

Proof of Theorem axprlem5OLD

Step	Hyp	Ref	Expression
1		ax-nul 5231	. 2 ⊢ ∃𝑠∀𝑛 ¬ 𝑛 ∈ 𝑠
2		nfa1 2164	. . . 4 ⊢ Ⅎ𝑠∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝)
3		nfv 1922	. . . 4 ⊢ Ⅎ𝑠 𝑤 = 𝑦
4	2, 3	nfan 1907	. . 3 ⊢ Ⅎ𝑠(∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)
5		pm2.21 123	. . . . . . . . 9 ⊢ (¬ 𝑛 ∈ 𝑠 → (𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
6	5	alimi 1819	. . . . . . . 8 ⊢ (∀𝑛 ¬ 𝑛 ∈ 𝑠 → ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
7	6	adantr 482	. . . . . . 7 ⊢ ((∀𝑛 ¬ 𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
8		df-ral 3056	. . . . . . 7 ⊢ (∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 ↔ ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
9	7, 8	sylibr 236	. . . . . 6 ⊢ ((∀𝑛 ¬ 𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → ∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛)
10		sp 2197	. . . . . . 7 ⊢ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) → (∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝))
11	10	ad2antrl 735	. . . . . 6 ⊢ ((∀𝑛 ¬ 𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → (∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝))
12	9, 11	mpd 15	. . . . 5 ⊢ ((∀𝑛 ¬ 𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → 𝑠 ∈ 𝑝)
13		simpl 484	. . . . . . 7 ⊢ ((∀𝑛 ¬ 𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → ∀𝑛 ¬ 𝑛 ∈ 𝑠)
14		alnex 1789	. . . . . . 7 ⊢ (∀𝑛 ¬ 𝑛 ∈ 𝑠 ↔ ¬ ∃𝑛 𝑛 ∈ 𝑠)
15	13, 14	sylib 220	. . . . . 6 ⊢ ((∀𝑛 ¬ 𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → ¬ ∃𝑛 𝑛 ∈ 𝑠)
16		simprr 779	. . . . . 6 ⊢ ((∀𝑛 ¬ 𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑦)
17		ifpfal 1082	. . . . . . 7 ⊢ (¬ ∃𝑛 𝑛 ∈ 𝑠 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑦))
18	17	biimpar 479	. . . . . 6 ⊢ ((¬ ∃𝑛 𝑛 ∈ 𝑠 ∧ 𝑤 = 𝑦) → if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))
19	15, 16, 18	syl2anc 591	. . . . 5 ⊢ ((∀𝑛 ¬ 𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))
20	12, 19	jca 517	. . . 4 ⊢ ((∀𝑛 ¬ 𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → (𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
21	20	expcom 415	. . 3 ⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦) → (∀𝑛 ¬ 𝑛 ∈ 𝑠 → (𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
22	4, 21	eximd 2230	. 2 ⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦) → (∃𝑠∀𝑛 ¬ 𝑛 ∈ 𝑠 → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
23	1, 22	mpi 20	1 ⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397  if-wif 1069  ∀wal 1546  ∃wex 1787  ∀wral 3055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-10 2154  ax-12 2191  ax-nul 5231

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-ifp 1070  df-tru 1551  df-ex 1788  df-nf 1792  df-ral 3056

This theorem is referenced by:  axprOLD  5364