Theorem mnuprdlem2 43017

Description: Lemma for mnuprd 43020. (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 2821	. . . . 5 ⊢ (𝑖 = {∅} → (𝑖 ∈ 𝑢 ↔ {∅} ∈ 𝑢))
2	1	anbi1d 630	. . . 4 ⊢ (𝑖 = {∅} → ((𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ ({∅} ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
3	2	rexbidv 3178	. . 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 5381	. . . . 5 ⊢ {∅} ∈ V
6	5	prid2 4766	. . . 4 ⊢ {∅} ∈ {∅, {∅}}
7	6	a1i 11	. . 3 ⊢ (𝜑 → {∅} ∈ {∅, {∅}})
8	3, 4, 7	rspcdva 3613	. 2 ⊢ (𝜑 → ∃𝑢 ∈ 𝐹 ({∅} ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))
9		simpl 483	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → 𝜑)
10		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → 𝑎 ∈ 𝐹)
11		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ {∅} ∈ 𝑎) → {∅} ∈ 𝑎)
12		0nep0 5355	. . . . . . . . . . . . . . 15 ⊢ ∅ ≠ {∅}
13	12	necomi 2995	. . . . . . . . . . . . . 14 ⊢ {∅} ≠ ∅
14	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {∅} ≠ ∅)
15		mnuprdlem2.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝐴 = ∅)
16		0ex 5306	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
17	16	sneqr 4840	. . . . . . . . . . . . . . . 16 ⊢ ({∅} = {𝐴} → ∅ = 𝐴)
18	17	eqcomd 2738	. . . . . . . . . . . . . . 15 ⊢ ({∅} = {𝐴} → 𝐴 = ∅)
19	15, 18	nsyl 140	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ {∅} = {𝐴})
20	19	neqned 2947	. . . . . . . . . . . . 13 ⊢ (𝜑 → {∅} ≠ {𝐴})
21	14, 20	nelprd 4658	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ {∅} ∈ {∅, {𝐴}})
22	21	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ {∅} ∈ 𝑎) → ¬ {∅} ∈ {∅, {𝐴}})
23	11, 22	elnelneqd 42939	. . . . . . . . . 10 ⊢ ((𝜑 ∧ {∅} ∈ 𝑎) → ¬ 𝑎 = {∅, {𝐴}})
24	23	adantrr 715	. . . . . . . . 9 ⊢ ((𝜑 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤)) → ¬ 𝑎 = {∅, {𝐴}})
25	24	adantrl 714	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ¬ 𝑎 = {∅, {𝐴}})
26		elpri 4649	. . . . . . . . . 10 ⊢ (𝑎 ∈ {{∅, {𝐴}}, {{∅}, {𝐵}}} → (𝑎 = {∅, {𝐴}} ∨ 𝑎 = {{∅}, {𝐵}}))
27		mnuprdlem2.1	. . . . . . . . . 10 ⊢ 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}
28	26, 27	eleq2s 2851	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐹 → (𝑎 = {∅, {𝐴}} ∨ 𝑎 = {{∅}, {𝐵}}))
29	28	ord 862	. . . . . . . 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 4921	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ∪ 𝑎 = ∪ {{∅}, {𝐵}})
32		snex 5430	. . . . . . . 8 ⊢ {𝐵} ∈ V
33	5, 32	unipr 4925	. . . . . . 7 ⊢ ∪ {{∅}, {𝐵}} = ({∅} ∪ {𝐵})
34		df-pr 4630	. . . . . . 7 ⊢ {∅, 𝐵} = ({∅} ∪ {𝐵})
35	33, 34	eqtr4i 2763	. . . . . 6 ⊢ ∪ {{∅}, {𝐵}} = {∅, 𝐵}
36	31, 35	eqtrdi 2788	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ∪ 𝑎 = {∅, 𝐵})
37		simprrr 780	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ∪ 𝑎 ⊆ 𝑤)
38	36, 37	eqsstrrd 4020	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → {∅, 𝐵} ⊆ 𝑤)
39		mnuprdlem2.4	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
40		prssg 4821	. . . . . 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 2817	. . 3 ⊢ (𝑢 = 𝑎 → ({∅} ∈ 𝑢 ↔ {∅} ∈ 𝑎))
46		unieq 4918	. . . 4 ⊢ (𝑢 = 𝑎 → ∪ 𝑢 = ∪ 𝑎)
47	46	sseq1d 4012	. . 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 42943	1 ⊢ (𝜑 → 𝐵 ∈ 𝑤)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ∪ cun 3945   ⊆ wss 3947  ∅c0 4321  {csn 4627  {cpr 4629  ∪ cuni 4907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-pw 4603  df-sn 4628  df-pr 4630  df-uni 4908

This theorem is referenced by:  mnuprdlem4  43019