Theorem axprlem4 5444

Description: Lemma for axpr 5446. The first 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
axprlem4	⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))

Distinct variable groups:   𝑥,𝑠   𝑤,𝑠   𝑡,𝑛,𝑠

Proof of Theorem axprlem4

Step	Hyp	Ref	Expression
1		axprlem1 5441	. . 3 ⊢ ∃𝑠∀𝑛(∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑛 ∈ 𝑠)
2	1	bm1.3ii 5320	. 2 ⊢ ∃𝑠∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛)
3		nfa1 2152	. . . 4 ⊢ Ⅎ𝑠∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝)
4		nfv 1913	. . . 4 ⊢ Ⅎ𝑠 𝑤 = 𝑥
5	3, 4	nfan 1898	. . 3 ⊢ Ⅎ𝑠(∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥)
6		biimp 215	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → (𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
7	6	alimi 1809	. . . . . . . 8 ⊢ (∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
8		df-ral 3068	. . . . . . . 8 ⊢ (∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 ↔ ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
9	7, 8	sylibr 234	. . . . . . 7 ⊢ (∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → ∀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	9, 10	mpan9 506	. . . . . 6 ⊢ ((∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) ∧ ∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝)) → 𝑠 ∈ 𝑝)
12	11	adantrr 716	. . . . 5 ⊢ ((∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥)) → 𝑠 ∈ 𝑝)
13		ax-nul 5324	. . . . . . 7 ⊢ ∃𝑛∀𝑡 ¬ 𝑡 ∈ 𝑛
14		nfa1 2152	. . . . . . . 8 ⊢ Ⅎ𝑛∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛)
15		sp 2184	. . . . . . . . 9 ⊢ (∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → (𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛))
16	15	biimprd 248	. . . . . . . 8 ⊢ (∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → (∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑛 ∈ 𝑠))
17	14, 16	eximd 2217	. . . . . . 7 ⊢ (∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → (∃𝑛∀𝑡 ¬ 𝑡 ∈ 𝑛 → ∃𝑛 𝑛 ∈ 𝑠))
18	13, 17	mpi 20	. . . . . 6 ⊢ (∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → ∃𝑛 𝑛 ∈ 𝑠)
19		simprr 772	. . . . . 6 ⊢ ((∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥)) → 𝑤 = 𝑥)
20		ifptru 1075	. . . . . . 7 ⊢ (∃𝑛 𝑛 ∈ 𝑠 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑥))
21	20	biimpar 477	. . . . . 6 ⊢ ((∃𝑛 𝑛 ∈ 𝑠 ∧ 𝑤 = 𝑥) → if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))
22	18, 19, 21	syl2an2r 684	. . . . 5 ⊢ ((∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥)) → if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))
23	12, 22	jca 511	. . . 4 ⊢ ((∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥)) → (𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
24	23	expcom 413	. . 3 ⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥) → (∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → (𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
25	5, 24	eximd 2217	. 2 ⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥) → (∃𝑠∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
26	2, 25	mpi 20	1 ⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ 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-9 2118  ax-10 2141  ax-12 2178  ax-sep 5317  ax-nul 5324  ax-pow 5383

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