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Theorem mnuprdlem1 43333
Description: Lemma for mnuprd 43337. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Hypotheses
Ref Expression
mnuprdlem1.1 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}
mnuprdlem1.3 (𝜑𝐴𝑈)
mnuprdlem1.4 (𝜑𝐵𝑈)
mnuprdlem1.8 (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))
Assertion
Ref Expression
mnuprdlem1 (𝜑𝐴𝑤)
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 mnuprdlem1
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2819 . . . . 5 (𝑖 = ∅ → (𝑖𝑢 ↔ ∅ ∈ 𝑢))
21anbi1d 628 . . . 4 (𝑖 = ∅ → ((𝑖𝑢 𝑢𝑤) ↔ (∅ ∈ 𝑢 𝑢𝑤)))
32rexbidv 3176 . . 3 (𝑖 = ∅ → (∃𝑢𝐹 (𝑖𝑢 𝑢𝑤) ↔ ∃𝑢𝐹 (∅ ∈ 𝑢 𝑢𝑤)))
4 mnuprdlem1.8 . . 3 (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))
5 0ex 5306 . . . . 5 ∅ ∈ V
65prid1 4765 . . . 4 ∅ ∈ {∅, {∅}}
76a1i 11 . . 3 (𝜑 → ∅ ∈ {∅, {∅}})
83, 4, 7rspcdva 3612 . 2 (𝜑 → ∃𝑢𝐹 (∅ ∈ 𝑢 𝑢𝑤))
9 mnuprdlem1.3 . . . 4 (𝜑𝐴𝑈)
109adantr 479 . . 3 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → 𝐴𝑈)
11 simprl 767 . . . . . . 7 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → 𝑎𝐹)
12 simpr 483 . . . . . . . . . 10 ((𝜑 ∧ ∅ ∈ 𝑎) → ∅ ∈ 𝑎)
13 0nep0 5355 . . . . . . . . . . . . 13 ∅ ≠ {∅}
1413a1i 11 . . . . . . . . . . . 12 (𝜑 → ∅ ≠ {∅})
15 mnuprdlem1.4 . . . . . . . . . . . . . 14 (𝜑𝐵𝑈)
1615snn0d 4778 . . . . . . . . . . . . 13 (𝜑 → {𝐵} ≠ ∅)
1716necomd 2994 . . . . . . . . . . . 12 (𝜑 → ∅ ≠ {𝐵})
1814, 17nelprd 4658 . . . . . . . . . . 11 (𝜑 → ¬ ∅ ∈ {{∅}, {𝐵}})
1918adantr 479 . . . . . . . . . 10 ((𝜑 ∧ ∅ ∈ 𝑎) → ¬ ∅ ∈ {{∅}, {𝐵}})
2012, 19elnelneqd 43256 . . . . . . . . 9 ((𝜑 ∧ ∅ ∈ 𝑎) → ¬ 𝑎 = {{∅}, {𝐵}})
2120adantrr 713 . . . . . . . 8 ((𝜑 ∧ (∅ ∈ 𝑎 𝑎𝑤)) → ¬ 𝑎 = {{∅}, {𝐵}})
2221adantrl 712 . . . . . . 7 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → ¬ 𝑎 = {{∅}, {𝐵}})
23 elpri 4649 . . . . . . . . . 10 (𝑎 ∈ {{∅, {𝐴}}, {{∅}, {𝐵}}} → (𝑎 = {∅, {𝐴}} ∨ 𝑎 = {{∅}, {𝐵}}))
24 mnuprdlem1.1 . . . . . . . . . 10 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}
2523, 24eleq2s 2849 . . . . . . . . 9 (𝑎𝐹 → (𝑎 = {∅, {𝐴}} ∨ 𝑎 = {{∅}, {𝐵}}))
2625orcomd 867 . . . . . . . 8 (𝑎𝐹 → (𝑎 = {{∅}, {𝐵}} ∨ 𝑎 = {∅, {𝐴}}))
2726ord 860 . . . . . . 7 (𝑎𝐹 → (¬ 𝑎 = {{∅}, {𝐵}} → 𝑎 = {∅, {𝐴}}))
2811, 22, 27sylc 65 . . . . . 6 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → 𝑎 = {∅, {𝐴}})
2928unieqd 4921 . . . . 5 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → 𝑎 = {∅, {𝐴}})
30 snex 5430 . . . . . . 7 {𝐴} ∈ V
315, 30unipr 4925 . . . . . 6 {∅, {𝐴}} = (∅ ∪ {𝐴})
32 uncom 4152 . . . . . 6 (∅ ∪ {𝐴}) = ({𝐴} ∪ ∅)
33 un0 4389 . . . . . 6 ({𝐴} ∪ ∅) = {𝐴}
3431, 32, 333eqtri 2762 . . . . 5 {∅, {𝐴}} = {𝐴}
3529, 34eqtrdi 2786 . . . 4 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → 𝑎 = {𝐴})
36 simprrr 778 . . . 4 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → 𝑎𝑤)
3735, 36eqsstrrd 4020 . . 3 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → {𝐴} ⊆ 𝑤)
38 snssg 4786 . . . 4 (𝐴𝑈 → (𝐴𝑤 ↔ {𝐴} ⊆ 𝑤))
3938biimprd 247 . . 3 (𝐴𝑈 → ({𝐴} ⊆ 𝑤𝐴𝑤))
4010, 37, 39sylc 65 . 2 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → 𝐴𝑤)
41 eleq2w 2815 . . 3 (𝑢 = 𝑎 → (∅ ∈ 𝑢 ↔ ∅ ∈ 𝑎))
42 unieq 4918 . . . 4 (𝑢 = 𝑎 𝑢 = 𝑎)
4342sseq1d 4012 . . 3 (𝑢 = 𝑎 → ( 𝑢𝑤 𝑎𝑤))
4441, 43anbi12d 629 . 2 (𝑢 = 𝑎 → ((∅ ∈ 𝑢 𝑢𝑤) ↔ (∅ ∈ 𝑎 𝑎𝑤)))
458, 40, 44rexlimddvcbvw 43260 1 (𝜑𝐴𝑤)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 843   = wceq 1539  wcel 2104  wne 2938  wral 3059  wrex 3068  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 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-sn 4628  df-pr 4630  df-uni 4908
This theorem is referenced by:  mnuprdlem4  43336
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