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Theorem mnuprdlem3 43033
Description: Lemma for mnuprd 43035. (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 4651 . . . . 5 (𝑖 ∈ {∅, {∅}} → (𝑖 = ∅ ∨ 𝑖 = {∅}))
3 0ex 5308 . . . . . . . . . 10 ∅ ∈ V
43prid1 4767 . . . . . . . . 9 ∅ ∈ {∅, {𝐴}}
54a1i 11 . . . . . . . 8 (((𝜑𝑖 = ∅) ∧ 𝑎 = {∅, {𝐴}}) → ∅ ∈ {∅, {𝐴}})
6 simplr 768 . . . . . . . 8 (((𝜑𝑖 = ∅) ∧ 𝑎 = {∅, {𝐴}}) → 𝑖 = ∅)
7 simpr 486 . . . . . . . 8 (((𝜑𝑖 = ∅) ∧ 𝑎 = {∅, {𝐴}}) → 𝑎 = {∅, {𝐴}})
85, 6, 73eltr4d 2849 . . . . . . 7 (((𝜑𝑖 = ∅) ∧ 𝑎 = {∅, {𝐴}}) → 𝑖𝑎)
9 prex 5433 . . . . . . . . . 10 {∅, {𝐴}} ∈ V
109prid1 4767 . . . . . . . . 9 {∅, {𝐴}} ∈ {{∅, {𝐴}}, {{∅}, {𝐵}}}
11 mnuprdlem3.1 . . . . . . . . 9 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}
1210, 11eleqtrri 2833 . . . . . . . 8 {∅, {𝐴}} ∈ 𝐹
1312a1i 11 . . . . . . 7 ((𝜑𝑖 = ∅) → {∅, {𝐴}} ∈ 𝐹)
148, 13rspcime 3617 . . . . . 6 ((𝜑𝑖 = ∅) → ∃𝑎𝐹 𝑖𝑎)
15 p0ex 5383 . . . . . . . . . 10 {∅} ∈ V
1615prid1 4767 . . . . . . . . 9 {∅} ∈ {{∅}, {𝐵}}
1716a1i 11 . . . . . . . 8 (((𝜑𝑖 = {∅}) ∧ 𝑎 = {{∅}, {𝐵}}) → {∅} ∈ {{∅}, {𝐵}})
18 simplr 768 . . . . . . . 8 (((𝜑𝑖 = {∅}) ∧ 𝑎 = {{∅}, {𝐵}}) → 𝑖 = {∅})
19 simpr 486 . . . . . . . 8 (((𝜑𝑖 = {∅}) ∧ 𝑎 = {{∅}, {𝐵}}) → 𝑎 = {{∅}, {𝐵}})
2017, 18, 193eltr4d 2849 . . . . . . 7 (((𝜑𝑖 = {∅}) ∧ 𝑎 = {{∅}, {𝐵}}) → 𝑖𝑎)
21 prex 5433 . . . . . . . . . 10 {{∅}, {𝐵}} ∈ V
2221prid2 4768 . . . . . . . . 9 {{∅}, {𝐵}} ∈ {{∅, {𝐴}}, {{∅}, {𝐵}}}
2322, 11eleqtrri 2833 . . . . . . . 8 {{∅}, {𝐵}} ∈ 𝐹
2423a1i 11 . . . . . . 7 ((𝜑𝑖 = {∅}) → {{∅}, {𝐵}} ∈ 𝐹)
2520, 24rspcime 3617 . . . . . 6 ((𝜑𝑖 = {∅}) → ∃𝑎𝐹 𝑖𝑎)
2614, 25jaodan 957 . . . . 5 ((𝜑 ∧ (𝑖 = ∅ ∨ 𝑖 = {∅})) → ∃𝑎𝐹 𝑖𝑎)
272, 26sylan2 594 . . . 4 ((𝜑𝑖 ∈ {∅, {∅}}) → ∃𝑎𝐹 𝑖𝑎)
28 elequ2 2122 . . . . 5 (𝑎 = 𝑣 → (𝑖𝑎𝑖𝑣))
2928cbvrexvw 3236 . . . 4 (∃𝑎𝐹 𝑖𝑎 ↔ ∃𝑣𝐹 𝑖𝑣)
3027, 29sylib 217 . . 3 ((𝜑𝑖 ∈ {∅, {∅}}) → ∃𝑣𝐹 𝑖𝑣)
3130ex 414 . 2 (𝜑 → (𝑖 ∈ {∅, {∅}} → ∃𝑣𝐹 𝑖𝑣))
321, 31ralrimi 3255 1 (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑣𝐹 𝑖𝑣)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846   = wceq 1542  wnf 1786  wcel 2107  wral 3062  wrex 3071  c0 4323  {csn 4629  {cpr 4631
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-pw 4605  df-sn 4630  df-pr 4632
This theorem is referenced by:  mnuprdlem4  43034
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