Theorem mnuprdlem2 41891

Description: Lemma for mnuprd 41894. (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 2826	. . . . 5 ⊢ (𝑖 = {∅} → (𝑖 ∈ 𝑢 ↔ {∅} ∈ 𝑢))
2	1	anbi1d 630	. . . 4 ⊢ (𝑖 = {∅} → ((𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ ({∅} ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
3	2	rexbidv 3226	. . 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		p0ex 5307	. . . . 5 ⊢ {∅} ∈ V
6	5	prid2 4699	. . . 4 ⊢ {∅} ∈ {∅, {∅}}
7	6	a1i 11	. . 3 ⊢ (𝜑 → {∅} ∈ {∅, {∅}})
8	3, 4, 7	rspcdva 3562	. 2 ⊢ (𝜑 → ∃𝑢 ∈ 𝐹 ({∅} ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))
9		simpl 483	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → 𝜑)
10		simprl 768	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → 𝑎 ∈ 𝐹)
11		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ {∅} ∈ 𝑎) → {∅} ∈ 𝑎)
12		0nep0 5280	. . . . . . . . . . . . . . 15 ⊢ ∅ ≠ {∅}
13	12	necomi 2998	. . . . . . . . . . . . . 14 ⊢ {∅} ≠ ∅
14	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {∅} ≠ ∅)
15		mnuprdlem2.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝐴 = ∅)
16		0ex 5231	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
17	16	sneqr 4771	. . . . . . . . . . . . . . . 16 ⊢ ({∅} = {𝐴} → ∅ = 𝐴)
18	17	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ ({∅} = {𝐴} → 𝐴 = ∅)
19	15, 18	nsyl 140	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ {∅} = {𝐴})
20	19	neqned 2950	. . . . . . . . . . . . 13 ⊢ (𝜑 → {∅} ≠ {𝐴})
21	14, 20	nelprd 4592	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ {∅} ∈ {∅, {𝐴}})
22	21	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ {∅} ∈ 𝑎) → ¬ {∅} ∈ {∅, {𝐴}})
23	11, 22	elnelneqd 41813	. . . . . . . . . 10 ⊢ ((𝜑 ∧ {∅} ∈ 𝑎) → ¬ 𝑎 = {∅, {𝐴}})
24	23	adantrr 714	. . . . . . . . 9 ⊢ ((𝜑 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤)) → ¬ 𝑎 = {∅, {𝐴}})
25	24	adantrl 713	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ¬ 𝑎 = {∅, {𝐴}})
26		elpri 4583	. . . . . . . . . 10 ⊢ (𝑎 ∈ {{∅, {𝐴}}, {{∅}, {𝐵}}} → (𝑎 = {∅, {𝐴}} ∨ 𝑎 = {{∅}, {𝐵}}))
27		mnuprdlem2.1	. . . . . . . . . 10 ⊢ 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}
28	26, 27	eleq2s 2857	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐹 → (𝑎 = {∅, {𝐴}} ∨ 𝑎 = {{∅}, {𝐵}}))
29	28	ord 861	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐹 → (¬ 𝑎 = {∅, {𝐴}} → 𝑎 = {{∅}, {𝐵}}))
30	10, 25, 29	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → 𝑎 = {{∅}, {𝐵}})
31	30	unieqd 4853	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ∪ 𝑎 = ∪ {{∅}, {𝐵}})
32		snex 5354	. . . . . . . 8 ⊢ {𝐵} ∈ V
33	5, 32	unipr 4857	. . . . . . 7 ⊢ ∪ {{∅}, {𝐵}} = ({∅} ∪ {𝐵})
34		df-pr 4564	. . . . . . 7 ⊢ {∅, 𝐵} = ({∅} ∪ {𝐵})
35	33, 34	eqtr4i 2769	. . . . . 6 ⊢ ∪ {{∅}, {𝐵}} = {∅, 𝐵}
36	31, 35	eqtrdi 2794	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ∪ 𝑎 = {∅, 𝐵})
37		simprrr 779	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ∪ 𝑎 ⊆ 𝑤)
38	36, 37	eqsstrrd 3960	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → {∅, 𝐵} ⊆ 𝑤)
39		mnuprdlem2.4	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
40		prssg 4752	. . . . . 6 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑈) → ((∅ ∈ 𝑤 ∧ 𝐵 ∈ 𝑤) ↔ {∅, 𝐵} ⊆ 𝑤))
41	16, 39, 40	sylancr 587	. . . . 5 ⊢ (𝜑 → ((∅ ∈ 𝑤 ∧ 𝐵 ∈ 𝑤) ↔ {∅, 𝐵} ⊆ 𝑤))
42	41	biimprd 247	. . . 4 ⊢ (𝜑 → ({∅, 𝐵} ⊆ 𝑤 → (∅ ∈ 𝑤 ∧ 𝐵 ∈ 𝑤)))
43	9, 38, 42	sylc 65	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → (∅ ∈ 𝑤 ∧ 𝐵 ∈ 𝑤))
44	43	simprd 496	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → 𝐵 ∈ 𝑤)
45		eleq2w 2822	. . 3 ⊢ (𝑢 = 𝑎 → ({∅} ∈ 𝑢 ↔ {∅} ∈ 𝑎))
46		unieq 4850	. . . 4 ⊢ (𝑢 = 𝑎 → ∪ 𝑢 = ∪ 𝑎)
47	46	sseq1d 3952	. . 3 ⊢ (𝑢 = 𝑎 → (∪ 𝑢 ⊆ 𝑤 ↔ ∪ 𝑎 ⊆ 𝑤))
48	45, 47	anbi12d 631	. 2 ⊢ (𝑢 = 𝑎 → (({∅} ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤)))
49	8, 44, 48	rexlimddvcbvw 41817	1 ⊢ (𝜑 → 𝐵 ∈ 𝑤)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  {csn 4561  {cpr 4563  ∪ cuni 4839

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-sn 4562  df-pr 4564  df-uni 4840

This theorem is referenced by:  mnuprdlem4  41893