Theorem axprlem5 5445

Description: Lemma for axpr 5446. The second element of the pair is included in any superset of the set whose existence is asserted by the axiom of replacement. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.)

Assertion

Ref	Expression
axprlem5	⊢ ((∀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 axprlem5

Step	Hyp	Ref	Expression
1		ax-nul 5324	. 2 ⊢ ∃𝑠∀𝑛 ¬ 𝑛 ∈ 𝑠
2		nfa1 2152	. . . 4 ⊢ Ⅎ𝑠∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝)
3		nfv 1913	. . . 4 ⊢ Ⅎ𝑠 𝑤 = 𝑦
4	2, 3	nfan 1898	. . 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 1809	. . . . . . . 8 ⊢ (∀𝑛 ¬ 𝑛 ∈ 𝑠 → ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
7	6	adantr 480	. . . . . . 7 ⊢ ((∀𝑛 ¬ 𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
8		df-ral 3068	. . . . . . 7 ⊢ (∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 ↔ ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
9	7, 8	sylibr 234	. . . . . 6 ⊢ ((∀𝑛 ¬ 𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → ∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛)
10		sp 2184	. . . . . . 7 ⊢ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) → (∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝))
11	10	ad2antrl 727	. . . . . 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 482	. . . . . . 7 ⊢ ((∀𝑛 ¬ 𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → ∀𝑛 ¬ 𝑛 ∈ 𝑠)
14		alnex 1779	. . . . . . 7 ⊢ (∀𝑛 ¬ 𝑛 ∈ 𝑠 ↔ ¬ ∃𝑛 𝑛 ∈ 𝑠)
15	13, 14	sylib 218	. . . . . 6 ⊢ ((∀𝑛 ¬ 𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → ¬ ∃𝑛 𝑛 ∈ 𝑠)
16		simprr 772	. . . . . 6 ⊢ ((∀𝑛 ¬ 𝑛 ∈ 𝑠 ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦)) → 𝑤 = 𝑦)
17		ifpfal 1076	. . . . . . 7 ⊢ (¬ ∃𝑛 𝑛 ∈ 𝑠 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑦))
18	17	biimpar 477	. . . . . 6 ⊢ ((¬ ∃𝑛 𝑛 ∈ 𝑠 ∧ 𝑤 = 𝑦) → if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))
19	15, 16, 18	syl2anc 583	. . . . 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 511	. . . 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 413	. . 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 2217	. 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 395  if-wif 1063  ∀wal 1535  ∃wex 1777  ∀wral 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2178  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ifp 1064  df-tru 1540  df-ex 1778  df-nf 1782  df-ral 3068

This theorem is referenced by:  axpr  5446