Theorem mnuprdlem2 41425

Description: Lemma for mnuprd 41428. (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 2820	. . . . 5 ⊢ (𝑖 = {∅} → (𝑖 ∈ 𝑢 ↔ {∅} ∈ 𝑢))
2	1	anbi1d 633	. . . 4 ⊢ (𝑖 = {∅} → ((𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ ({∅} ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
3	2	rexbidv 3207	. . 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 5252	. . . . 5 ⊢ {∅} ∈ V
6	5	prid2 4655	. . . 4 ⊢ {∅} ∈ {∅, {∅}}
7	6	a1i 11	. . 3 ⊢ (𝜑 → {∅} ∈ {∅, {∅}})
8	3, 4, 7	rspcdva 3529	. 2 ⊢ (𝜑 → ∃𝑢 ∈ 𝐹 ({∅} ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))
9		simpl 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → 𝜑)
10		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → 𝑎 ∈ 𝐹)
11		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ {∅} ∈ 𝑎) → {∅} ∈ 𝑎)
12		0nep0 5225	. . . . . . . . . . . . . . 15 ⊢ ∅ ≠ {∅}
13	12	necomi 2988	. . . . . . . . . . . . . 14 ⊢ {∅} ≠ ∅
14	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {∅} ≠ ∅)
15		mnuprdlem2.5	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝐴 = ∅)
16		0ex 5176	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ∈ V
17	16	sneqr 4727	. . . . . . . . . . . . . . . 16 ⊢ ({∅} = {𝐴} → ∅ = 𝐴)
18	17	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ ({∅} = {𝐴} → 𝐴 = ∅)
19	15, 18	nsyl 142	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ {∅} = {𝐴})
20	19	neqned 2941	. . . . . . . . . . . . 13 ⊢ (𝜑 → {∅} ≠ {𝐴})
21	14, 20	nelprd 4548	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ {∅} ∈ {∅, {𝐴}})
22	21	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ {∅} ∈ 𝑎) → ¬ {∅} ∈ {∅, {𝐴}})
23	11, 22	elnelneqd 41352	. . . . . . . . . 10 ⊢ ((𝜑 ∧ {∅} ∈ 𝑎) → ¬ 𝑎 = {∅, {𝐴}})
24	23	adantrr 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤)) → ¬ 𝑎 = {∅, {𝐴}})
25	24	adantrl 716	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ¬ 𝑎 = {∅, {𝐴}})
26		elpri 4539	. . . . . . . . . 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 863	. . . . . . . 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 4811	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ∪ 𝑎 = ∪ {{∅}, {𝐵}})
32		snex 5299	. . . . . . . 8 ⊢ {𝐵} ∈ V
33	5, 32	unipr 4815	. . . . . . 7 ⊢ ∪ {{∅}, {𝐵}} = ({∅} ∪ {𝐵})
34		df-pr 4520	. . . . . . 7 ⊢ {∅, 𝐵} = ({∅} ∪ {𝐵})
35	33, 34	eqtr4i 2764	. . . . . 6 ⊢ ∪ {{∅}, {𝐵}} = {∅, 𝐵}
36	31, 35	eqtrdi 2789	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ∪ 𝑎 = {∅, 𝐵})
37		simprrr 782	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → ∪ 𝑎 ⊆ 𝑤)
38	36, 37	eqsstrrd 3917	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → {∅, 𝐵} ⊆ 𝑤)
39		mnuprdlem2.4	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
40		prssg 4708	. . . . . 6 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑈) → ((∅ ∈ 𝑤 ∧ 𝐵 ∈ 𝑤) ↔ {∅, 𝐵} ⊆ 𝑤))
41	16, 39, 40	sylancr 590	. . . . 5 ⊢ (𝜑 → ((∅ ∈ 𝑤 ∧ 𝐵 ∈ 𝑤) ↔ {∅, 𝐵} ⊆ 𝑤))
42	41	biimprd 251	. . . 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 499	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐹 ∧ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤))) → 𝐵 ∈ 𝑤)
45		eleq2w 2816	. . 3 ⊢ (𝑢 = 𝑎 → ({∅} ∈ 𝑢 ↔ {∅} ∈ 𝑎))
46		unieq 4808	. . . 4 ⊢ (𝑢 = 𝑎 → ∪ 𝑢 = ∪ 𝑎)
47	46	sseq1d 3909	. . 3 ⊢ (𝑢 = 𝑎 → (∪ 𝑢 ⊆ 𝑤 ↔ ∪ 𝑎 ⊆ 𝑤))
48	45, 47	anbi12d 634	. 2 ⊢ (𝑢 = 𝑎 → (({∅} ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ ({∅} ∈ 𝑎 ∧ ∪ 𝑎 ⊆ 𝑤)))
49	8, 44, 48	rexlimddvcbvw 41356	1 ⊢ (𝜑 → 𝐵 ∈ 𝑤)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 846   = wceq 1542   ∈ wcel 2113   ≠ wne 2934  ∀wral 3053  ∃wrex 3054  Vcvv 3398   ∪ cun 3842   ⊆ wss 3844  ∅c0 4212  {csn 4517  {cpr 4519  ∪ cuni 4797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-pw 4491  df-sn 4518  df-pr 4520  df-uni 4798

This theorem is referenced by:  mnuprdlem4  41427