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Theorem prsal 41041
Description: The pair of the empty set and the whole base is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
prsal (𝑋𝑉 → {∅, 𝑋} ∈ SAlg)

Proof of Theorem prsal
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 0ex 4942 . . . . 5 ∅ ∈ V
21prid1 4441 . . . 4 ∅ ∈ {∅, 𝑋}
32a1i 11 . . 3 (𝑋𝑉 → ∅ ∈ {∅, 𝑋})
41a1i 11 . . . . . . . . 9 (𝑋𝑉 → ∅ ∈ V)
5 id 22 . . . . . . . . 9 (𝑋𝑉𝑋𝑉)
6 uniprg 4602 . . . . . . . . 9 ((∅ ∈ V ∧ 𝑋𝑉) → {∅, 𝑋} = (∅ ∪ 𝑋))
74, 5, 6syl2anc 696 . . . . . . . 8 (𝑋𝑉 {∅, 𝑋} = (∅ ∪ 𝑋))
8 uncom 3900 . . . . . . . . . 10 (∅ ∪ 𝑋) = (𝑋 ∪ ∅)
9 un0 4110 . . . . . . . . . 10 (𝑋 ∪ ∅) = 𝑋
108, 9eqtri 2782 . . . . . . . . 9 (∅ ∪ 𝑋) = 𝑋
1110a1i 11 . . . . . . . 8 (𝑋𝑉 → (∅ ∪ 𝑋) = 𝑋)
12 eqidd 2761 . . . . . . . 8 (𝑋𝑉𝑋 = 𝑋)
137, 11, 123eqtrd 2798 . . . . . . 7 (𝑋𝑉 {∅, 𝑋} = 𝑋)
1413difeq1d 3870 . . . . . 6 (𝑋𝑉 → ( {∅, 𝑋} ∖ 𝑦) = (𝑋𝑦))
1514adantr 472 . . . . 5 ((𝑋𝑉𝑦 ∈ {∅, 𝑋}) → ( {∅, 𝑋} ∖ 𝑦) = (𝑋𝑦))
16 difeq2 3865 . . . . . . . . . 10 (𝑦 = ∅ → (𝑋𝑦) = (𝑋 ∖ ∅))
1716adantl 473 . . . . . . . . 9 ((𝑋𝑉𝑦 = ∅) → (𝑋𝑦) = (𝑋 ∖ ∅))
18 dif0 4093 . . . . . . . . . 10 (𝑋 ∖ ∅) = 𝑋
1918a1i 11 . . . . . . . . 9 ((𝑋𝑉𝑦 = ∅) → (𝑋 ∖ ∅) = 𝑋)
2017, 19eqtrd 2794 . . . . . . . 8 ((𝑋𝑉𝑦 = ∅) → (𝑋𝑦) = 𝑋)
21 prid2g 4440 . . . . . . . . 9 (𝑋𝑉𝑋 ∈ {∅, 𝑋})
2221adantr 472 . . . . . . . 8 ((𝑋𝑉𝑦 = ∅) → 𝑋 ∈ {∅, 𝑋})
2320, 22eqeltrd 2839 . . . . . . 7 ((𝑋𝑉𝑦 = ∅) → (𝑋𝑦) ∈ {∅, 𝑋})
2423adantlr 753 . . . . . 6 (((𝑋𝑉𝑦 ∈ {∅, 𝑋}) ∧ 𝑦 = ∅) → (𝑋𝑦) ∈ {∅, 𝑋})
25 simpll 807 . . . . . . 7 (((𝑋𝑉𝑦 ∈ {∅, 𝑋}) ∧ ¬ 𝑦 = ∅) → 𝑋𝑉)
26 simpl 474 . . . . . . . . 9 ((𝑦 ∈ {∅, 𝑋} ∧ ¬ 𝑦 = ∅) → 𝑦 ∈ {∅, 𝑋})
27 neqne 2940 . . . . . . . . . 10 𝑦 = ∅ → 𝑦 ≠ ∅)
2827adantl 473 . . . . . . . . 9 ((𝑦 ∈ {∅, 𝑋} ∧ ¬ 𝑦 = ∅) → 𝑦 ≠ ∅)
29 elprn1 40368 . . . . . . . . 9 ((𝑦 ∈ {∅, 𝑋} ∧ 𝑦 ≠ ∅) → 𝑦 = 𝑋)
3026, 28, 29syl2anc 696 . . . . . . . 8 ((𝑦 ∈ {∅, 𝑋} ∧ ¬ 𝑦 = ∅) → 𝑦 = 𝑋)
3130adantll 752 . . . . . . 7 (((𝑋𝑉𝑦 ∈ {∅, 𝑋}) ∧ ¬ 𝑦 = ∅) → 𝑦 = 𝑋)
32 difeq2 3865 . . . . . . . . . 10 (𝑦 = 𝑋 → (𝑋𝑦) = (𝑋𝑋))
33 difid 4091 . . . . . . . . . . 11 (𝑋𝑋) = ∅
3433a1i 11 . . . . . . . . . 10 (𝑦 = 𝑋 → (𝑋𝑋) = ∅)
3532, 34eqtrd 2794 . . . . . . . . 9 (𝑦 = 𝑋 → (𝑋𝑦) = ∅)
362a1i 11 . . . . . . . . 9 (𝑦 = 𝑋 → ∅ ∈ {∅, 𝑋})
3735, 36eqeltrd 2839 . . . . . . . 8 (𝑦 = 𝑋 → (𝑋𝑦) ∈ {∅, 𝑋})
3837adantl 473 . . . . . . 7 ((𝑋𝑉𝑦 = 𝑋) → (𝑋𝑦) ∈ {∅, 𝑋})
3925, 31, 38syl2anc 696 . . . . . 6 (((𝑋𝑉𝑦 ∈ {∅, 𝑋}) ∧ ¬ 𝑦 = ∅) → (𝑋𝑦) ∈ {∅, 𝑋})
4024, 39pm2.61dan 867 . . . . 5 ((𝑋𝑉𝑦 ∈ {∅, 𝑋}) → (𝑋𝑦) ∈ {∅, 𝑋})
4115, 40eqeltrd 2839 . . . 4 ((𝑋𝑉𝑦 ∈ {∅, 𝑋}) → ( {∅, 𝑋} ∖ 𝑦) ∈ {∅, 𝑋})
4241ralrimiva 3104 . . 3 (𝑋𝑉 → ∀𝑦 ∈ {∅, 𝑋} ( {∅, 𝑋} ∖ 𝑦) ∈ {∅, 𝑋})
43 elpwi 4312 . . . . . . . . . . . . 13 (𝑦 ∈ 𝒫 {∅, 𝑋} → 𝑦 ⊆ {∅, 𝑋})
44 uniss 4610 . . . . . . . . . . . . 13 (𝑦 ⊆ {∅, 𝑋} → 𝑦 {∅, 𝑋})
4543, 44syl 17 . . . . . . . . . . . 12 (𝑦 ∈ 𝒫 {∅, 𝑋} → 𝑦 {∅, 𝑋})
4645adantl 473 . . . . . . . . . . 11 ((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) → 𝑦 {∅, 𝑋})
4713adantr 472 . . . . . . . . . . 11 ((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) → {∅, 𝑋} = 𝑋)
4846, 47sseqtrd 3782 . . . . . . . . . 10 ((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) → 𝑦𝑋)
4948adantr 472 . . . . . . . . 9 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋𝑦) → 𝑦𝑋)
50 elssuni 4619 . . . . . . . . . 10 (𝑋𝑦𝑋 𝑦)
5150adantl 473 . . . . . . . . 9 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋𝑦) → 𝑋 𝑦)
5249, 51jca 555 . . . . . . . 8 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋𝑦) → ( 𝑦𝑋𝑋 𝑦))
53 eqss 3759 . . . . . . . 8 ( 𝑦 = 𝑋 ↔ ( 𝑦𝑋𝑋 𝑦))
5452, 53sylibr 224 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋𝑦) → 𝑦 = 𝑋)
5521ad2antrr 764 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋𝑦) → 𝑋 ∈ {∅, 𝑋})
5654, 55eqeltrd 2839 . . . . . 6 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋𝑦) → 𝑦 ∈ {∅, 𝑋})
57 id 22 . . . . . . . . . . . 12 (𝑦 ∈ 𝒫 {∅, 𝑋} → 𝑦 ∈ 𝒫 {∅, 𝑋})
58 pwpr 4582 . . . . . . . . . . . 12 𝒫 {∅, 𝑋} = ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}})
5957, 58syl6eleq 2849 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 {∅, 𝑋} → 𝑦 ∈ ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}}))
6059adantr 472 . . . . . . . . . 10 ((𝑦 ∈ 𝒫 {∅, 𝑋} ∧ ¬ 𝑋𝑦) → 𝑦 ∈ ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}}))
6160adantll 752 . . . . . . . . 9 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋𝑦) → 𝑦 ∈ ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}}))
62 snidg 4351 . . . . . . . . . . . . . . . 16 (𝑋𝑉𝑋 ∈ {𝑋})
6362adantr 472 . . . . . . . . . . . . . . 15 ((𝑋𝑉𝑦 = {𝑋}) → 𝑋 ∈ {𝑋})
64 id 22 . . . . . . . . . . . . . . . . 17 (𝑦 = {𝑋} → 𝑦 = {𝑋})
6564eqcomd 2766 . . . . . . . . . . . . . . . 16 (𝑦 = {𝑋} → {𝑋} = 𝑦)
6665adantl 473 . . . . . . . . . . . . . . 15 ((𝑋𝑉𝑦 = {𝑋}) → {𝑋} = 𝑦)
6763, 66eleqtrd 2841 . . . . . . . . . . . . . 14 ((𝑋𝑉𝑦 = {𝑋}) → 𝑋𝑦)
6867adantlr 753 . . . . . . . . . . . . 13 (((𝑋𝑉𝑦 ∈ {{𝑋}, {∅, 𝑋}}) ∧ 𝑦 = {𝑋}) → 𝑋𝑦)
695ad2antrr 764 . . . . . . . . . . . . . 14 (((𝑋𝑉𝑦 ∈ {{𝑋}, {∅, 𝑋}}) ∧ ¬ 𝑦 = {𝑋}) → 𝑋𝑉)
70 simpl 474 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ {{𝑋}, {∅, 𝑋}} ∧ ¬ 𝑦 = {𝑋}) → 𝑦 ∈ {{𝑋}, {∅, 𝑋}})
7164necon3bi 2958 . . . . . . . . . . . . . . . . 17 𝑦 = {𝑋} → 𝑦 ≠ {𝑋})
7271adantl 473 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ {{𝑋}, {∅, 𝑋}} ∧ ¬ 𝑦 = {𝑋}) → 𝑦 ≠ {𝑋})
73 elprn1 40368 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ {{𝑋}, {∅, 𝑋}} ∧ 𝑦 ≠ {𝑋}) → 𝑦 = {∅, 𝑋})
7470, 72, 73syl2anc 696 . . . . . . . . . . . . . . 15 ((𝑦 ∈ {{𝑋}, {∅, 𝑋}} ∧ ¬ 𝑦 = {𝑋}) → 𝑦 = {∅, 𝑋})
7574adantll 752 . . . . . . . . . . . . . 14 (((𝑋𝑉𝑦 ∈ {{𝑋}, {∅, 𝑋}}) ∧ ¬ 𝑦 = {𝑋}) → 𝑦 = {∅, 𝑋})
7621adantr 472 . . . . . . . . . . . . . . 15 ((𝑋𝑉𝑦 = {∅, 𝑋}) → 𝑋 ∈ {∅, 𝑋})
77 id 22 . . . . . . . . . . . . . . . . 17 (𝑦 = {∅, 𝑋} → 𝑦 = {∅, 𝑋})
7877eqcomd 2766 . . . . . . . . . . . . . . . 16 (𝑦 = {∅, 𝑋} → {∅, 𝑋} = 𝑦)
7978adantl 473 . . . . . . . . . . . . . . 15 ((𝑋𝑉𝑦 = {∅, 𝑋}) → {∅, 𝑋} = 𝑦)
8076, 79eleqtrd 2841 . . . . . . . . . . . . . 14 ((𝑋𝑉𝑦 = {∅, 𝑋}) → 𝑋𝑦)
8169, 75, 80syl2anc 696 . . . . . . . . . . . . 13 (((𝑋𝑉𝑦 ∈ {{𝑋}, {∅, 𝑋}}) ∧ ¬ 𝑦 = {𝑋}) → 𝑋𝑦)
8268, 81pm2.61dan 867 . . . . . . . . . . . 12 ((𝑋𝑉𝑦 ∈ {{𝑋}, {∅, 𝑋}}) → 𝑋𝑦)
8382adantlr 753 . . . . . . . . . . 11 (((𝑋𝑉 ∧ ¬ 𝑋𝑦) ∧ 𝑦 ∈ {{𝑋}, {∅, 𝑋}}) → 𝑋𝑦)
84 simplr 809 . . . . . . . . . . 11 (((𝑋𝑉 ∧ ¬ 𝑋𝑦) ∧ 𝑦 ∈ {{𝑋}, {∅, 𝑋}}) → ¬ 𝑋𝑦)
8583, 84pm2.65da 601 . . . . . . . . . 10 ((𝑋𝑉 ∧ ¬ 𝑋𝑦) → ¬ 𝑦 ∈ {{𝑋}, {∅, 𝑋}})
8685adantlr 753 . . . . . . . . 9 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋𝑦) → ¬ 𝑦 ∈ {{𝑋}, {∅, 𝑋}})
87 elunnel2 39697 . . . . . . . . 9 ((𝑦 ∈ ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}}) ∧ ¬ 𝑦 ∈ {{𝑋}, {∅, 𝑋}}) → 𝑦 ∈ {∅, {∅}})
8861, 86, 87syl2anc 696 . . . . . . . 8 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋𝑦) → 𝑦 ∈ {∅, {∅}})
89 unieq 4596 . . . . . . . . . . 11 (𝑦 = ∅ → 𝑦 = ∅)
90 uni0 4617 . . . . . . . . . . . 12 ∅ = ∅
9190a1i 11 . . . . . . . . . . 11 (𝑦 = ∅ → ∅ = ∅)
9289, 91eqtrd 2794 . . . . . . . . . 10 (𝑦 = ∅ → 𝑦 = ∅)
9392adantl 473 . . . . . . . . 9 ((𝑦 ∈ {∅, {∅}} ∧ 𝑦 = ∅) → 𝑦 = ∅)
94 simpl 474 . . . . . . . . . . 11 ((𝑦 ∈ {∅, {∅}} ∧ ¬ 𝑦 = ∅) → 𝑦 ∈ {∅, {∅}})
9527adantl 473 . . . . . . . . . . 11 ((𝑦 ∈ {∅, {∅}} ∧ ¬ 𝑦 = ∅) → 𝑦 ≠ ∅)
96 elprn1 40368 . . . . . . . . . . 11 ((𝑦 ∈ {∅, {∅}} ∧ 𝑦 ≠ ∅) → 𝑦 = {∅})
9794, 95, 96syl2anc 696 . . . . . . . . . 10 ((𝑦 ∈ {∅, {∅}} ∧ ¬ 𝑦 = ∅) → 𝑦 = {∅})
98 unieq 4596 . . . . . . . . . . 11 (𝑦 = {∅} → 𝑦 = {∅})
991unisn 4603 . . . . . . . . . . . 12 {∅} = ∅
10099a1i 11 . . . . . . . . . . 11 (𝑦 = {∅} → {∅} = ∅)
10198, 100eqtrd 2794 . . . . . . . . . 10 (𝑦 = {∅} → 𝑦 = ∅)
10297, 101syl 17 . . . . . . . . 9 ((𝑦 ∈ {∅, {∅}} ∧ ¬ 𝑦 = ∅) → 𝑦 = ∅)
10393, 102pm2.61dan 867 . . . . . . . 8 (𝑦 ∈ {∅, {∅}} → 𝑦 = ∅)
10488, 103syl 17 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋𝑦) → 𝑦 = ∅)
1052a1i 11 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋𝑦) → ∅ ∈ {∅, 𝑋})
106104, 105eqeltrd 2839 . . . . . 6 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋𝑦) → 𝑦 ∈ {∅, 𝑋})
10756, 106pm2.61dan 867 . . . . 5 ((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) → 𝑦 ∈ {∅, 𝑋})
108107a1d 25 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) → (𝑦 ≼ ω → 𝑦 ∈ {∅, 𝑋}))
109108ralrimiva 3104 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 {∅, 𝑋} (𝑦 ≼ ω → 𝑦 ∈ {∅, 𝑋}))
1103, 42, 1093jca 1123 . 2 (𝑋𝑉 → (∅ ∈ {∅, 𝑋} ∧ ∀𝑦 ∈ {∅, 𝑋} ( {∅, 𝑋} ∖ 𝑦) ∈ {∅, 𝑋} ∧ ∀𝑦 ∈ 𝒫 {∅, 𝑋} (𝑦 ≼ ω → 𝑦 ∈ {∅, 𝑋})))
111 prex 5058 . . . 4 {∅, 𝑋} ∈ V
112111a1i 11 . . 3 (𝑋𝑉 → {∅, 𝑋} ∈ V)
113 issal 41037 . . 3 ({∅, 𝑋} ∈ V → ({∅, 𝑋} ∈ SAlg ↔ (∅ ∈ {∅, 𝑋} ∧ ∀𝑦 ∈ {∅, 𝑋} ( {∅, 𝑋} ∖ 𝑦) ∈ {∅, 𝑋} ∧ ∀𝑦 ∈ 𝒫 {∅, 𝑋} (𝑦 ≼ ω → 𝑦 ∈ {∅, 𝑋}))))
114112, 113syl 17 . 2 (𝑋𝑉 → ({∅, 𝑋} ∈ SAlg ↔ (∅ ∈ {∅, 𝑋} ∧ ∀𝑦 ∈ {∅, 𝑋} ( {∅, 𝑋} ∖ 𝑦) ∈ {∅, 𝑋} ∧ ∀𝑦 ∈ 𝒫 {∅, 𝑋} (𝑦 ≼ ω → 𝑦 ∈ {∅, 𝑋}))))
115110, 114mpbird 247 1 (𝑋𝑉 → {∅, 𝑋} ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  Vcvv 3340  cdif 3712  cun 3713  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321  {cpr 4323   cuni 4588   class class class wbr 4804  ωcom 7230  cdom 8119  SAlgcsalg 41031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-pr 4324  df-uni 4589  df-salg 41032
This theorem is referenced by: (None)
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