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Theorem mnuprdlem3 40107
Description: Lemma for mnuprd 40109. (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 4494 . . . . 5 (𝑖 ∈ {∅, {∅}} → (𝑖 = ∅ ∨ 𝑖 = {∅}))
3 0ex 5102 . . . . . . . . . 10 ∅ ∈ V
43prid1 4605 . . . . . . . . 9 ∅ ∈ {∅, {𝐴}}
54a1i 11 . . . . . . . 8 (((𝜑𝑖 = ∅) ∧ 𝑎 = {∅, {𝐴}}) → ∅ ∈ {∅, {𝐴}})
6 simplr 765 . . . . . . . 8 (((𝜑𝑖 = ∅) ∧ 𝑎 = {∅, {𝐴}}) → 𝑖 = ∅)
7 simpr 485 . . . . . . . 8 (((𝜑𝑖 = ∅) ∧ 𝑎 = {∅, {𝐴}}) → 𝑎 = {∅, {𝐴}})
85, 6, 73eltr4d 2898 . . . . . . 7 (((𝜑𝑖 = ∅) ∧ 𝑎 = {∅, {𝐴}}) → 𝑖𝑎)
9 prex 5224 . . . . . . . . . 10 {∅, {𝐴}} ∈ V
109prid1 4605 . . . . . . . . 9 {∅, {𝐴}} ∈ {{∅, {𝐴}}, {{∅}, {𝐵}}}
11 mnuprdlem3.1 . . . . . . . . 9 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}
1210, 11eleqtrri 2882 . . . . . . . 8 {∅, {𝐴}} ∈ 𝐹
1312a1i 11 . . . . . . 7 ((𝜑𝑖 = ∅) → {∅, {𝐴}} ∈ 𝐹)
148, 13rr-rspce 40045 . . . . . 6 ((𝜑𝑖 = ∅) → ∃𝑎𝐹 𝑖𝑎)
15 p0ex 5175 . . . . . . . . . 10 {∅} ∈ V
1615prid1 4605 . . . . . . . . 9 {∅} ∈ {{∅}, {𝐵}}
1716a1i 11 . . . . . . . 8 (((𝜑𝑖 = {∅}) ∧ 𝑎 = {{∅}, {𝐵}}) → {∅} ∈ {{∅}, {𝐵}})
18 simplr 765 . . . . . . . 8 (((𝜑𝑖 = {∅}) ∧ 𝑎 = {{∅}, {𝐵}}) → 𝑖 = {∅})
19 simpr 485 . . . . . . . 8 (((𝜑𝑖 = {∅}) ∧ 𝑎 = {{∅}, {𝐵}}) → 𝑎 = {{∅}, {𝐵}})
2017, 18, 193eltr4d 2898 . . . . . . 7 (((𝜑𝑖 = {∅}) ∧ 𝑎 = {{∅}, {𝐵}}) → 𝑖𝑎)
21 prex 5224 . . . . . . . . . 10 {{∅}, {𝐵}} ∈ V
2221prid2 4606 . . . . . . . . 9 {{∅}, {𝐵}} ∈ {{∅, {𝐴}}, {{∅}, {𝐵}}}
2322, 11eleqtrri 2882 . . . . . . . 8 {{∅}, {𝐵}} ∈ 𝐹
2423a1i 11 . . . . . . 7 ((𝜑𝑖 = {∅}) → {{∅}, {𝐵}} ∈ 𝐹)
2520, 24rr-rspce 40045 . . . . . 6 ((𝜑𝑖 = {∅}) → ∃𝑎𝐹 𝑖𝑎)
2614, 25jaodan 952 . . . . 5 ((𝜑 ∧ (𝑖 = ∅ ∨ 𝑖 = {∅})) → ∃𝑎𝐹 𝑖𝑎)
272, 26sylan2 592 . . . 4 ((𝜑𝑖 ∈ {∅, {∅}}) → ∃𝑎𝐹 𝑖𝑎)
28 elequ2 2096 . . . . 5 (𝑎 = 𝑣 → (𝑖𝑎𝑖𝑣))
2928cbvrexv 3404 . . . 4 (∃𝑎𝐹 𝑖𝑎 ↔ ∃𝑣𝐹 𝑖𝑣)
3027, 29sylib 219 . . 3 ((𝜑𝑖 ∈ {∅, {∅}}) → ∃𝑣𝐹 𝑖𝑣)
3130ex 413 . 2 (𝜑 → (𝑖 ∈ {∅, {∅}} → ∃𝑣𝐹 𝑖𝑣))
321, 31ralrimi 3183 1 (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑣𝐹 𝑖𝑣)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 842   = wceq 1522  wnf 1765  wcel 2081  wral 3105  wrex 3106  c0 4211  {csn 4472  {cpr 4474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-pw 4455  df-sn 4473  df-pr 4475
This theorem is referenced by:  mnuprdlem4  40108
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