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Theorem mnuprdlem1 40918
Description: Lemma for mnuprd 40922. (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 2901 . . . . 5 (𝑖 = ∅ → (𝑖𝑢 ↔ ∅ ∈ 𝑢))
21anbi1d 632 . . . 4 (𝑖 = ∅ → ((𝑖𝑢 𝑢𝑤) ↔ (∅ ∈ 𝑢 𝑢𝑤)))
32rexbidv 3283 . . 3 (𝑖 = ∅ → (∃𝑢𝐹 (𝑖𝑢 𝑢𝑤) ↔ ∃𝑢𝐹 (∅ ∈ 𝑢 𝑢𝑤)))
4 mnuprdlem1.8 . . 3 (𝜑 → ∀𝑖 ∈ {∅, {∅}}∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))
5 0ex 5187 . . . . 5 ∅ ∈ V
65prid1 4672 . . . 4 ∅ ∈ {∅, {∅}}
76a1i 11 . . 3 (𝜑 → ∅ ∈ {∅, {∅}})
83, 4, 7rspcdva 3600 . 2 (𝜑 → ∃𝑢𝐹 (∅ ∈ 𝑢 𝑢𝑤))
9 mnuprdlem1.3 . . . 4 (𝜑𝐴𝑈)
109adantr 484 . . 3 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → 𝐴𝑈)
11 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → 𝑎𝐹)
12 simpr 488 . . . . . . . . . 10 ((𝜑 ∧ ∅ ∈ 𝑎) → ∅ ∈ 𝑎)
13 0nep0 5235 . . . . . . . . . . . . 13 ∅ ≠ {∅}
1413a1i 11 . . . . . . . . . . . 12 (𝜑 → ∅ ≠ {∅})
15 mnuprdlem1.4 . . . . . . . . . . . . . 14 (𝜑𝐵𝑈)
16 snnzg 4684 . . . . . . . . . . . . . 14 (𝐵𝑈 → {𝐵} ≠ ∅)
1715, 16syl 17 . . . . . . . . . . . . 13 (𝜑 → {𝐵} ≠ ∅)
1817necomd 3066 . . . . . . . . . . . 12 (𝜑 → ∅ ≠ {𝐵})
1914, 18nelprd 4570 . . . . . . . . . . 11 (𝜑 → ¬ ∅ ∈ {{∅}, {𝐵}})
2019adantr 484 . . . . . . . . . 10 ((𝜑 ∧ ∅ ∈ 𝑎) → ¬ ∅ ∈ {{∅}, {𝐵}})
2112, 20elnelneqd 40845 . . . . . . . . 9 ((𝜑 ∧ ∅ ∈ 𝑎) → ¬ 𝑎 = {{∅}, {𝐵}})
2221adantrr 716 . . . . . . . 8 ((𝜑 ∧ (∅ ∈ 𝑎 𝑎𝑤)) → ¬ 𝑎 = {{∅}, {𝐵}})
2322adantrl 715 . . . . . . 7 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → ¬ 𝑎 = {{∅}, {𝐵}})
24 elpri 4561 . . . . . . . . . 10 (𝑎 ∈ {{∅, {𝐴}}, {{∅}, {𝐵}}} → (𝑎 = {∅, {𝐴}} ∨ 𝑎 = {{∅}, {𝐵}}))
25 mnuprdlem1.1 . . . . . . . . . 10 𝐹 = {{∅, {𝐴}}, {{∅}, {𝐵}}}
2624, 25eleq2s 2932 . . . . . . . . 9 (𝑎𝐹 → (𝑎 = {∅, {𝐴}} ∨ 𝑎 = {{∅}, {𝐵}}))
2726orcomd 868 . . . . . . . 8 (𝑎𝐹 → (𝑎 = {{∅}, {𝐵}} ∨ 𝑎 = {∅, {𝐴}}))
2827ord 861 . . . . . . 7 (𝑎𝐹 → (¬ 𝑎 = {{∅}, {𝐵}} → 𝑎 = {∅, {𝐴}}))
2911, 23, 28sylc 65 . . . . . 6 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → 𝑎 = {∅, {𝐴}})
3029unieqd 4827 . . . . 5 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → 𝑎 = {∅, {𝐴}})
31 snex 5309 . . . . . . 7 {𝐴} ∈ V
325, 31unipr 4830 . . . . . 6 {∅, {𝐴}} = (∅ ∪ {𝐴})
33 uncom 4104 . . . . . 6 (∅ ∪ {𝐴}) = ({𝐴} ∪ ∅)
34 un0 4316 . . . . . 6 ({𝐴} ∪ ∅) = {𝐴}
3532, 33, 343eqtri 2849 . . . . 5 {∅, {𝐴}} = {𝐴}
3630, 35syl6eq 2873 . . . 4 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → 𝑎 = {𝐴})
37 simprrr 781 . . . 4 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → 𝑎𝑤)
3836, 37eqsstrrd 3981 . . 3 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → {𝐴} ⊆ 𝑤)
39 snssg 4691 . . . 4 (𝐴𝑈 → (𝐴𝑤 ↔ {𝐴} ⊆ 𝑤))
4039biimprd 251 . . 3 (𝐴𝑈 → ({𝐴} ⊆ 𝑤𝐴𝑤))
4110, 38, 40sylc 65 . 2 ((𝜑 ∧ (𝑎𝐹 ∧ (∅ ∈ 𝑎 𝑎𝑤))) → 𝐴𝑤)
42 eleq2w 2897 . . 3 (𝑢 = 𝑎 → (∅ ∈ 𝑢 ↔ ∅ ∈ 𝑎))
43 unieq 4824 . . . 4 (𝑢 = 𝑎 𝑢 = 𝑎)
4443sseq1d 3973 . . 3 (𝑢 = 𝑎 → ( 𝑢𝑤 𝑎𝑤))
4542, 44anbi12d 633 . 2 (𝑢 = 𝑎 → ((∅ ∈ 𝑢 𝑢𝑤) ↔ (∅ ∈ 𝑎 𝑎𝑤)))
468, 41, 45rexlimddvcbvw 40849 1 (𝜑𝐴𝑤)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844   = wceq 1538  wcel 2114  wne 3011  wral 3130  wrex 3131  cun 3906  wss 3908  c0 4265  {csn 4539  {cpr 4541   cuni 4813
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 2116  ax-9 2124  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540  df-pr 4542  df-uni 4814
This theorem is referenced by:  mnuprdlem4  40921
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