Theorem mnuprdlem2 45024

Description: Lemma for mnuprd 45027. (Contributed by Rohan Ridenour, 11-Aug-2023.)

Hypotheses

Ref	Expression
mnuprdlem2.1	⊢ 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}
mnuprdlem2.4	⊢ (𝜑 → 𝐵 ∈ 𝑈)
mnuprdlem2.5	⊢ (𝜑 → ¬ 𝐴 = ∅)
mnuprdlem2.8	⊢ (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))

Assertion

Ref	Expression
mnuprdlem2	⊢ (𝜑 → 𝐵 ∈ 𝑤)

Distinct variable groups:   𝑤,𝑖,𝑢   𝑢,𝐹,𝑖

Allowed substitution hints:   𝜑(𝑤, 𝑢, 𝑖)   𝐴(𝑤, 𝑢, 𝑖)   𝐵(𝑤, 𝑢, 𝑖)   𝑈(𝑤, 𝑢, 𝑖)   𝐹(𝑤)

Proof of Theorem mnuprdlem2

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2854	. . . . 5 ⊢ (𝑖 = {∅} → (𝑖 ∈ 𝑢 ↔ {∅} ∈ 𝑢))
2	1	anbi1d 643	. . . 4 ⊢ (𝑖 = {∅} → ((𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ ({∅} ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
3	2	rexbidv 3192	. . 3 ⊢ (𝑖 = {∅} → (∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ ∃𝑢 ∈ 𝐹 ({∅} ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
4		mnuprdlem2.8	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))
5		snex 5415	. . . . 5 ⊢ {∅} ∈ V
6	5	prid2 4734	. . . 4 ⊢ {∅} ∈ {∅, {∅}}
7	6	a1i 11	. . 3 ⊢ (𝜑 → {∅} ∈ {∅, {∅}})
8	3, 4, 7	rspcdva 3585	. 2 ⊢ (𝜑 → ∃𝑢 ∈ 𝐹 ({∅} ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))
9		simpl 488	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → 𝜑)
10		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → 𝑎 ∈ 𝐹)
11		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ {∅} ∈ 𝑎) → {∅} ∈ 𝑎)
12		0nep0 5333	. . . . . . . . . . . . . . 15 ⊢ ∅ ≠ {∅}
13	12	necomi 3015	. . . . . . . . . . . . . 14 ⊢ {∅} ≠ ∅
14	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {∅} ≠ ∅)
15		mnuprdlem2.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝐴 = ∅)
16		0ex 5275	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
17	16	sneqr 4810	. . . . . . . . . . . . . . . 16 ⊢ ({∅} = {𝐴} → ∅ = 𝐴)
18	17	eqcomd 2772	. . . . . . . . . . . . . . 15 ⊢ ({∅} = {𝐴} → 𝐴 = ∅)
19	15, 18	nsyl 141	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ {∅} = {𝐴})
20	19	neqned 2968	. . . . . . . . . . . . 13 ⊢ (𝜑 → {∅} ≠ {𝐴})
21	14, 20	nelprd 4628	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ {∅} ∈ {∅, {𝐴}})
22	21	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ {∅} ∈ 𝑎) → ¬ {∅} ∈ {∅, {𝐴}})
23	11, 22	elnelneqd 3060	. . . . . . . . . 10 ⊢ ((𝜑 ∧ {∅} ∈ 𝑎) → ¬ 𝑎 = {∅, {𝐴}})
24	23	adantrr 730	. . . . . . . . 9 ⊢ ((𝜑 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤)) → ¬ 𝑎 = {∅, {𝐴}})
25	24	adantrl 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ¬ 𝑎 = {∅, {𝐴}})
26		elpri 4618	. . . . . . . . . 10 ⊢ (𝑎 ∈ {{∅, {𝐴}}, {{∅}, {𝐵}}} → (𝑎 = {∅, {𝐴}} ∨ 𝑎 = {{∅}, {𝐵}}))
27		mnuprdlem2.1	. . . . . . . . . 10 ⊢ 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}
28	26, 27	eleq2s 2884	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐹 → (𝑎 = {∅, {𝐴}} ∨ 𝑎 = {{∅}, {𝐵}}))
29	28	ord 878	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐹 → (¬ 𝑎 = {∅, {𝐴}} → 𝑎 = {{∅}, {𝐵}}))
30	10, 25, 29	sylc 66	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → 𝑎 = {{∅}, {𝐵}})
31	30	unieqd 4890	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ∪ 𝑎 = ∪ {{∅}, {𝐵}})
32		snex 5415	. . . . . . . 8 ⊢ {𝐵} ∈ V
33	5, 32	unipr 4894	. . . . . . 7 ⊢ ∪ {{∅}, {𝐵}} = ({∅} ∪ {𝐵})
34		df-pr 4597	. . . . . . 7 ⊢ {∅, 𝐵} = ({∅} ∪ {𝐵})
35	33, 34	eqtr4i 2792	. . . . . 6 ⊢ ∪ {{∅}, {𝐵}} = {∅, 𝐵}
36	31, 35	eqtrdi 2817	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ∪ 𝑎 = {∅, 𝐵})
37		simprrr 794	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ∪ 𝑎 ⊆ 𝑤)
38	36, 37	eqsstrrd 3975	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → {∅, 𝐵} ⊆ 𝑤)
39		mnuprdlem2.4	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
40		prssg 4790	. . . . . 6 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑈) → ((∅ ∈ 𝑤 ∧ 𝐵 ∈ 𝑤) ↔ {∅, 𝐵} ⊆ 𝑤))
41	16, 39, 40	sylancr 599	. . . . 5 ⊢ (𝜑 → ((∅ ∈ 𝑤 ∧ 𝐵 ∈ 𝑤) ↔ {∅, 𝐵} ⊆ 𝑤))
42	41	biimprd 251	. . . 4 ⊢ (𝜑 → ({∅, 𝐵} ⊆ 𝑤 → (∅ ∈ 𝑤 ∧ 𝐵 ∈ 𝑤)))
43	9, 38, 42	sylc 66	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → (∅ ∈ 𝑤 ∧ 𝐵 ∈ 𝑤))
44	43	simprd 501	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → 𝐵 ∈ 𝑤)
45		eleq2w 2850	. . 3 ⊢ (𝑢 = 𝑎 → ({∅} ∈ 𝑢 ↔ {∅} ∈ 𝑎))
46		unieq 4888	. . . 4 ⊢ (𝑢 = 𝑎 → ∪ 𝑢 = ∪ 𝑎)
47	46	sseq1d 3971	. . 3 ⊢ (𝑢 = 𝑎 → (∪ 𝑢 ⊆ 𝑤 ↔ ∪ 𝑎 ⊆ 𝑤))
48	45, 47	anbi12d 644	. 2 ⊢ (𝑢 = 𝑎 → (({∅} ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤)))
49	8, 44, 48	rexlimddvcbvw 44971	1 ⊢ (𝜑 → 𝐵 ∈ 𝑤)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082  ∃wrex 3092  Vcvv 3458   ∪ cun 3906   ⊆ wss 3908  ∅c0 4289  {csn 4594  {cpr 4596  ∪ cuni 4877

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-sn 4595  df-pr 4597  df-uni 4878

This theorem is used by:  mnuprdlem4  45026