Theorem axprlem4 5422

Description: Lemma for axpr 5424. 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 5419	. . 3 ⊢ ∃𝑠∀𝑛(∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑛 ∈ 𝑠)
2	1	bm1.3ii 5299	. 2 ⊢ ∃𝑠∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛)
3		nfa1 2141	. . . 4 ⊢ Ⅎ𝑠∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝)
4		nfv 1910	. . . 4 ⊢ Ⅎ𝑠 𝑤 = 𝑥
5	3, 4	nfan 1895	. . 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 1806	. . . . . . . 8 ⊢ (∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) → ∀𝑛(𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))
8		df-ral 3052	. . . . . . . 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 2172	. . . . . . 7 ⊢ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) → (∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝))
11	9, 10	mpan9 505	. . . . . 6 ⊢ ((∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) ∧ ∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝)) → 𝑠 ∈ 𝑝)
12	11	adantrr 715	. . . . 5 ⊢ ((∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥)) → 𝑠 ∈ 𝑝)
13		ax-nul 5303	. . . . . . 7 ⊢ ∃𝑛∀𝑡 ¬ 𝑡 ∈ 𝑛
14		nfa1 2141	. . . . . . . 8 ⊢ Ⅎ𝑛∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛)
15		sp 2172	. . . . . . . . 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 2205	. . . . . . 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 771	. . . . . 6 ⊢ ((∀𝑛(𝑛 ∈ 𝑠 ↔ ∀𝑡 ¬ 𝑡 ∈ 𝑛) ∧ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥)) → 𝑤 = 𝑥)
20		ifptru 1072	. . . . . . 7 ⊢ (∃𝑛 𝑛 ∈ 𝑠 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑥))
21	20	biimpar 476	. . . . . 6 ⊢ ((∃𝑛 𝑛 ∈ 𝑠 ∧ 𝑤 = 𝑥) → if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))
22	18, 19, 21	syl2an2r 683	. . . . 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 510	. . . 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 412	. . 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 2205	. 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 394  if-wif 1060  ∀wal 1532  ∃wex 1774  ∀wral 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-10 2130  ax-12 2167  ax-sep 5296  ax-nul 5303  ax-pow 5361

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-tru 1537  df-ex 1775  df-nf 1779  df-ral 3052

This theorem is referenced by:  axpr  5424