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Theorem mnuprdlem3 41025
 Description: Lemma for mnuprd 41027. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Hypotheses
Ref Expression
mnuprdlem3.1 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}
mnuprdlem3.9 𝑖𝜑
Assertion
Ref Expression
mnuprdlem3 (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑣𝐹 𝑖𝑣)
Distinct variable groups:   𝑣,𝑖   𝑣,𝐹
Allowed substitution hints:   𝜑(𝑣,𝑖)   𝐴(𝑣,𝑖)   𝐵(𝑣,𝑖)   𝐹(𝑖)

Proof of Theorem mnuprdlem3
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 mnuprdlem3.9 . 2 𝑖𝜑
2 elpri 4547 . . . . 5 (𝑖 ∈ {∅, {∅}} → (𝑖 = ∅ ∨ 𝑖 = {∅}))
3 0ex 5176 . . . . . . . . . 10 ∅ ∈ V
43prid1 4658 . . . . . . . . 9 ∅ ∈ {∅, {𝐴}}
54a1i 11 . . . . . . . 8 (((𝜑𝑖 = ∅) ∧ 𝑎 = {∅, {𝐴}}) → ∅ ∈ {∅, {𝐴}})
6 simplr 768 . . . . . . . 8 (((𝜑𝑖 = ∅) ∧ 𝑎 = {∅, {𝐴}}) → 𝑖 = ∅)
7 simpr 488 . . . . . . . 8 (((𝜑𝑖 = ∅) ∧ 𝑎 = {∅, {𝐴}}) → 𝑎 = {∅, {𝐴}})
85, 6, 73eltr4d 2905 . . . . . . 7 (((𝜑𝑖 = ∅) ∧ 𝑎 = {∅, {𝐴}}) → 𝑖𝑎)
9 prex 5299 . . . . . . . . . 10 {∅, {𝐴}} ∈ V
109prid1 4658 . . . . . . . . 9 {∅, {𝐴}} ∈ {{∅, {𝐴}}, {{∅}, {𝐵}}}
11 mnuprdlem3.1 . . . . . . . . 9 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}
1210, 11eleqtrri 2889 . . . . . . . 8 {∅, {𝐴}} ∈ 𝐹
1312a1i 11 . . . . . . 7 ((𝜑𝑖 = ∅) → {∅, {𝐴}} ∈ 𝐹)
148, 13rspcime 3575 . . . . . 6 ((𝜑𝑖 = ∅) → ∃𝑎𝐹 𝑖𝑎)
15 p0ex 5251 . . . . . . . . . 10 {∅} ∈ V
1615prid1 4658 . . . . . . . . 9 {∅} ∈ {{∅}, {𝐵}}
1716a1i 11 . . . . . . . 8 (((𝜑𝑖 = {∅}) ∧ 𝑎 = {{∅}, {𝐵}}) → {∅} ∈ {{∅}, {𝐵}})
18 simplr 768 . . . . . . . 8 (((𝜑𝑖 = {∅}) ∧ 𝑎 = {{∅}, {𝐵}}) → 𝑖 = {∅})
19 simpr 488 . . . . . . . 8 (((𝜑𝑖 = {∅}) ∧ 𝑎 = {{∅}, {𝐵}}) → 𝑎 = {{∅}, {𝐵}})
2017, 18, 193eltr4d 2905 . . . . . . 7 (((𝜑𝑖 = {∅}) ∧ 𝑎 = {{∅}, {𝐵}}) → 𝑖𝑎)
21 prex 5299 . . . . . . . . . 10 {{∅}, {𝐵}} ∈ V
2221prid2 4659 . . . . . . . . 9 {{∅}, {𝐵}} ∈ {{∅, {𝐴}}, {{∅}, {𝐵}}}
2322, 11eleqtrri 2889 . . . . . . . 8 {{∅}, {𝐵}} ∈ 𝐹
2423a1i 11 . . . . . . 7 ((𝜑𝑖 = {∅}) → {{∅}, {𝐵}} ∈ 𝐹)
2520, 24rspcime 3575 . . . . . 6 ((𝜑𝑖 = {∅}) → ∃𝑎𝐹 𝑖𝑎)
2614, 25jaodan 955 . . . . 5 ((𝜑 ∧ (𝑖 = ∅ ∨ 𝑖 = {∅})) → ∃𝑎𝐹 𝑖𝑎)
272, 26sylan2 595 . . . 4 ((𝜑𝑖 ∈ {∅, {∅}}) → ∃𝑎𝐹 𝑖𝑎)
28 elequ2 2126 . . . . 5 (𝑎 = 𝑣 → (𝑖𝑎𝑖𝑣))
2928cbvrexvw 3397 . . . 4 (∃𝑎𝐹 𝑖𝑎 ↔ ∃𝑣𝐹 𝑖𝑣)
3027, 29sylib 221 . . 3 ((𝜑𝑖 ∈ {∅, {∅}}) → ∃𝑣𝐹 𝑖𝑣)
3130ex 416 . 2 (𝜑 → (𝑖 ∈ {∅, {∅}} → ∃𝑣𝐹 𝑖𝑣))
321, 31ralrimi 3180 1 (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑣𝐹 𝑖𝑣)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  ∅c0 4243  {csn 4525  {cpr 4527 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-pw 4499  df-sn 4526  df-pr 4528 This theorem is referenced by:  mnuprdlem4  41026
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