Theorem axprlem4 5425

Description: Lemma for axpr 5427. 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 5422	. . 3 ⊢ ∃𝑠∀𝑛(∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑛 ∈ 𝑠)
2	1	bm1.3ii 5303	. 2 ⊢ ∃𝑠∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛)
3		nfa1 2149	. . . 4 ⊢ Ⅎ𝑠∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝)
4		nfv 1918	. . . 4 ⊢ Ⅎ𝑠 𝑤 = 𝑥
5	3, 4	nfan 1903	. . 3 ⊢ Ⅎ𝑠(∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥)
6		biimp 214	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → (𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
7	6	alimi 1814	. . . . . . . 8 ⊢ (∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
8		df-ral 3063	. . . . . . . 8 ⊢ (∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 ↔ ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
9	7, 8	sylibr 233	. . . . . . 7 ⊢ (∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → ∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛)
10		sp 2177	. . . . . . 7 ⊢ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) → (∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝))
11	9, 10	mpan9 508	. . . . . 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 5307	. . . . . . 7 ⊢ ∃𝑛∀𝑡 ¬ 𝑡 ∈ 𝑛
14		nfa1 2149	. . . . . . . 8 ⊢ Ⅎ𝑛∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛)
15		sp 2177	. . . . . . . . 9 ⊢ (∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → (𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛))
16	15	biimprd 247	. . . . . . . 8 ⊢ (∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → (∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑛 ∈ 𝑠))
17	14, 16	eximd 2210	. . . . . . 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 479	. . . . . 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 513	. . . 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 415	. . 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 2210	. 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 205   ∧ wa 397  if-wif 1062  ∀wal 1540  ∃wex 1782  ∀wral 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-10 2138  ax-12 2172  ax-sep 5300  ax-nul 5307  ax-pow 5364

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-tru 1545  df-ex 1783  df-nf 1787  df-ral 3063

This theorem is referenced by:  axpr  5427