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Theorem prsal 39871
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 4755 . . . . 5 ∅ ∈ V
21prid1 4272 . . . 4 ∅ ∈ {∅, 𝑋}
32a1i 11 . . 3 (𝑋𝑉 → ∅ ∈ {∅, 𝑋})
41a1i 11 . . . . . . . . 9 (𝑋𝑉 → ∅ ∈ V)
5 id 22 . . . . . . . . 9 (𝑋𝑉𝑋𝑉)
6 uniprg 4421 . . . . . . . . 9 ((∅ ∈ V ∧ 𝑋𝑉) → {∅, 𝑋} = (∅ ∪ 𝑋))
74, 5, 6syl2anc 692 . . . . . . . 8 (𝑋𝑉 {∅, 𝑋} = (∅ ∪ 𝑋))
8 uncom 3740 . . . . . . . . . 10 (∅ ∪ 𝑋) = (𝑋 ∪ ∅)
9 un0 3944 . . . . . . . . . 10 (𝑋 ∪ ∅) = 𝑋
108, 9eqtri 2643 . . . . . . . . 9 (∅ ∪ 𝑋) = 𝑋
1110a1i 11 . . . . . . . 8 (𝑋𝑉 → (∅ ∪ 𝑋) = 𝑋)
12 eqidd 2622 . . . . . . . 8 (𝑋𝑉𝑋 = 𝑋)
137, 11, 123eqtrd 2659 . . . . . . 7 (𝑋𝑉 {∅, 𝑋} = 𝑋)
1413difeq1d 3710 . . . . . 6 (𝑋𝑉 → ( {∅, 𝑋} ∖ 𝑦) = (𝑋𝑦))
1514adantr 481 . . . . 5 ((𝑋𝑉𝑦 ∈ {∅, 𝑋}) → ( {∅, 𝑋} ∖ 𝑦) = (𝑋𝑦))
16 difeq2 3705 . . . . . . . . . 10 (𝑦 = ∅ → (𝑋𝑦) = (𝑋 ∖ ∅))
1716adantl 482 . . . . . . . . 9 ((𝑋𝑉𝑦 = ∅) → (𝑋𝑦) = (𝑋 ∖ ∅))
18 dif0 3929 . . . . . . . . . 10 (𝑋 ∖ ∅) = 𝑋
1918a1i 11 . . . . . . . . 9 ((𝑋𝑉𝑦 = ∅) → (𝑋 ∖ ∅) = 𝑋)
2017, 19eqtrd 2655 . . . . . . . 8 ((𝑋𝑉𝑦 = ∅) → (𝑋𝑦) = 𝑋)
21 prid2g 4271 . . . . . . . . 9 (𝑋𝑉𝑋 ∈ {∅, 𝑋})
2221adantr 481 . . . . . . . 8 ((𝑋𝑉𝑦 = ∅) → 𝑋 ∈ {∅, 𝑋})
2320, 22eqeltrd 2698 . . . . . . 7 ((𝑋𝑉𝑦 = ∅) → (𝑋𝑦) ∈ {∅, 𝑋})
2423adantlr 750 . . . . . 6 (((𝑋𝑉𝑦 ∈ {∅, 𝑋}) ∧ 𝑦 = ∅) → (𝑋𝑦) ∈ {∅, 𝑋})
25 simpll 789 . . . . . . 7 (((𝑋𝑉𝑦 ∈ {∅, 𝑋}) ∧ ¬ 𝑦 = ∅) → 𝑋𝑉)
26 simpl 473 . . . . . . . . 9 ((𝑦 ∈ {∅, 𝑋} ∧ ¬ 𝑦 = ∅) → 𝑦 ∈ {∅, 𝑋})
27 neqne 2798 . . . . . . . . . 10 𝑦 = ∅ → 𝑦 ≠ ∅)
2827adantl 482 . . . . . . . . 9 ((𝑦 ∈ {∅, 𝑋} ∧ ¬ 𝑦 = ∅) → 𝑦 ≠ ∅)
29 elprn1 39297 . . . . . . . . 9 ((𝑦 ∈ {∅, 𝑋} ∧ 𝑦 ≠ ∅) → 𝑦 = 𝑋)
3026, 28, 29syl2anc 692 . . . . . . . 8 ((𝑦 ∈ {∅, 𝑋} ∧ ¬ 𝑦 = ∅) → 𝑦 = 𝑋)
3130adantll 749 . . . . . . 7 (((𝑋𝑉𝑦 ∈ {∅, 𝑋}) ∧ ¬ 𝑦 = ∅) → 𝑦 = 𝑋)
32 difeq2 3705 . . . . . . . . . 10 (𝑦 = 𝑋 → (𝑋𝑦) = (𝑋𝑋))
33 difid 3927 . . . . . . . . . . 11 (𝑋𝑋) = ∅
3433a1i 11 . . . . . . . . . 10 (𝑦 = 𝑋 → (𝑋𝑋) = ∅)
3532, 34eqtrd 2655 . . . . . . . . 9 (𝑦 = 𝑋 → (𝑋𝑦) = ∅)
362a1i 11 . . . . . . . . 9 (𝑦 = 𝑋 → ∅ ∈ {∅, 𝑋})
3735, 36eqeltrd 2698 . . . . . . . 8 (𝑦 = 𝑋 → (𝑋𝑦) ∈ {∅, 𝑋})
3837adantl 482 . . . . . . 7 ((𝑋𝑉𝑦 = 𝑋) → (𝑋𝑦) ∈ {∅, 𝑋})
3925, 31, 38syl2anc 692 . . . . . 6 (((𝑋𝑉𝑦 ∈ {∅, 𝑋}) ∧ ¬ 𝑦 = ∅) → (𝑋𝑦) ∈ {∅, 𝑋})
4024, 39pm2.61dan 831 . . . . 5 ((𝑋𝑉𝑦 ∈ {∅, 𝑋}) → (𝑋𝑦) ∈ {∅, 𝑋})
4115, 40eqeltrd 2698 . . . 4 ((𝑋𝑉𝑦 ∈ {∅, 𝑋}) → ( {∅, 𝑋} ∖ 𝑦) ∈ {∅, 𝑋})
4241ralrimiva 2961 . . 3 (𝑋𝑉 → ∀𝑦 ∈ {∅, 𝑋} ( {∅, 𝑋} ∖ 𝑦) ∈ {∅, 𝑋})
43 elpwi 4145 . . . . . . . . . . . . 13 (𝑦 ∈ 𝒫 {∅, 𝑋} → 𝑦 ⊆ {∅, 𝑋})
44 uniss 4429 . . . . . . . . . . . . 13 (𝑦 ⊆ {∅, 𝑋} → 𝑦 {∅, 𝑋})
4543, 44syl 17 . . . . . . . . . . . 12 (𝑦 ∈ 𝒫 {∅, 𝑋} → 𝑦 {∅, 𝑋})
4645adantl 482 . . . . . . . . . . 11 ((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) → 𝑦 {∅, 𝑋})
4713adantr 481 . . . . . . . . . . 11 ((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) → {∅, 𝑋} = 𝑋)
4846, 47sseqtrd 3625 . . . . . . . . . 10 ((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) → 𝑦𝑋)
4948adantr 481 . . . . . . . . 9 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋𝑦) → 𝑦𝑋)
50 elssuni 4438 . . . . . . . . . 10 (𝑋𝑦𝑋 𝑦)
5150adantl 482 . . . . . . . . 9 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋𝑦) → 𝑋 𝑦)
5249, 51jca 554 . . . . . . . 8 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋𝑦) → ( 𝑦𝑋𝑋 𝑦))
53 eqss 3602 . . . . . . . 8 ( 𝑦 = 𝑋 ↔ ( 𝑦𝑋𝑋 𝑦))
5452, 53sylibr 224 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋𝑦) → 𝑦 = 𝑋)
5521ad2antrr 761 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋𝑦) → 𝑋 ∈ {∅, 𝑋})
5654, 55eqeltrd 2698 . . . . . 6 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋𝑦) → 𝑦 ∈ {∅, 𝑋})
57 id 22 . . . . . . . . . . . 12 (𝑦 ∈ 𝒫 {∅, 𝑋} → 𝑦 ∈ 𝒫 {∅, 𝑋})
58 pwpr 4403 . . . . . . . . . . . 12 𝒫 {∅, 𝑋} = ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}})
5957, 58syl6eleq 2708 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 {∅, 𝑋} → 𝑦 ∈ ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}}))
6059adantr 481 . . . . . . . . . 10 ((𝑦 ∈ 𝒫 {∅, 𝑋} ∧ ¬ 𝑋𝑦) → 𝑦 ∈ ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}}))
6160adantll 749 . . . . . . . . 9 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋𝑦) → 𝑦 ∈ ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}}))
62 snidg 4182 . . . . . . . . . . . . . . . 16 (𝑋𝑉𝑋 ∈ {𝑋})
6362adantr 481 . . . . . . . . . . . . . . 15 ((𝑋𝑉𝑦 = {𝑋}) → 𝑋 ∈ {𝑋})
64 id 22 . . . . . . . . . . . . . . . . 17 (𝑦 = {𝑋} → 𝑦 = {𝑋})
6564eqcomd 2627 . . . . . . . . . . . . . . . 16 (𝑦 = {𝑋} → {𝑋} = 𝑦)
6665adantl 482 . . . . . . . . . . . . . . 15 ((𝑋𝑉𝑦 = {𝑋}) → {𝑋} = 𝑦)
6763, 66eleqtrd 2700 . . . . . . . . . . . . . 14 ((𝑋𝑉𝑦 = {𝑋}) → 𝑋𝑦)
6867adantlr 750 . . . . . . . . . . . . 13 (((𝑋𝑉𝑦 ∈ {{𝑋}, {∅, 𝑋}}) ∧ 𝑦 = {𝑋}) → 𝑋𝑦)
695ad2antrr 761 . . . . . . . . . . . . . 14 (((𝑋𝑉𝑦 ∈ {{𝑋}, {∅, 𝑋}}) ∧ ¬ 𝑦 = {𝑋}) → 𝑋𝑉)
70 simpl 473 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ {{𝑋}, {∅, 𝑋}} ∧ ¬ 𝑦 = {𝑋}) → 𝑦 ∈ {{𝑋}, {∅, 𝑋}})
7164necon3bi 2816 . . . . . . . . . . . . . . . . 17 𝑦 = {𝑋} → 𝑦 ≠ {𝑋})
7271adantl 482 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ {{𝑋}, {∅, 𝑋}} ∧ ¬ 𝑦 = {𝑋}) → 𝑦 ≠ {𝑋})
73 elprn1 39297 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ {{𝑋}, {∅, 𝑋}} ∧ 𝑦 ≠ {𝑋}) → 𝑦 = {∅, 𝑋})
7470, 72, 73syl2anc 692 . . . . . . . . . . . . . . 15 ((𝑦 ∈ {{𝑋}, {∅, 𝑋}} ∧ ¬ 𝑦 = {𝑋}) → 𝑦 = {∅, 𝑋})
7574adantll 749 . . . . . . . . . . . . . 14 (((𝑋𝑉𝑦 ∈ {{𝑋}, {∅, 𝑋}}) ∧ ¬ 𝑦 = {𝑋}) → 𝑦 = {∅, 𝑋})
7621adantr 481 . . . . . . . . . . . . . . 15 ((𝑋𝑉𝑦 = {∅, 𝑋}) → 𝑋 ∈ {∅, 𝑋})
77 id 22 . . . . . . . . . . . . . . . . 17 (𝑦 = {∅, 𝑋} → 𝑦 = {∅, 𝑋})
7877eqcomd 2627 . . . . . . . . . . . . . . . 16 (𝑦 = {∅, 𝑋} → {∅, 𝑋} = 𝑦)
7978adantl 482 . . . . . . . . . . . . . . 15 ((𝑋𝑉𝑦 = {∅, 𝑋}) → {∅, 𝑋} = 𝑦)
8076, 79eleqtrd 2700 . . . . . . . . . . . . . 14 ((𝑋𝑉𝑦 = {∅, 𝑋}) → 𝑋𝑦)
8169, 75, 80syl2anc 692 . . . . . . . . . . . . 13 (((𝑋𝑉𝑦 ∈ {{𝑋}, {∅, 𝑋}}) ∧ ¬ 𝑦 = {𝑋}) → 𝑋𝑦)
8268, 81pm2.61dan 831 . . . . . . . . . . . 12 ((𝑋𝑉𝑦 ∈ {{𝑋}, {∅, 𝑋}}) → 𝑋𝑦)
8382adantlr 750 . . . . . . . . . . 11 (((𝑋𝑉 ∧ ¬ 𝑋𝑦) ∧ 𝑦 ∈ {{𝑋}, {∅, 𝑋}}) → 𝑋𝑦)
84 simplr 791 . . . . . . . . . . 11 (((𝑋𝑉 ∧ ¬ 𝑋𝑦) ∧ 𝑦 ∈ {{𝑋}, {∅, 𝑋}}) → ¬ 𝑋𝑦)
8583, 84pm2.65da 599 . . . . . . . . . 10 ((𝑋𝑉 ∧ ¬ 𝑋𝑦) → ¬ 𝑦 ∈ {{𝑋}, {∅, 𝑋}})
8685adantlr 750 . . . . . . . . 9 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋𝑦) → ¬ 𝑦 ∈ {{𝑋}, {∅, 𝑋}})
87 elunnel2 38716 . . . . . . . . 9 ((𝑦 ∈ ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}}) ∧ ¬ 𝑦 ∈ {{𝑋}, {∅, 𝑋}}) → 𝑦 ∈ {∅, {∅}})
8861, 86, 87syl2anc 692 . . . . . . . 8 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋𝑦) → 𝑦 ∈ {∅, {∅}})
89 unieq 4415 . . . . . . . . . . 11 (𝑦 = ∅ → 𝑦 = ∅)
90 uni0 4436 . . . . . . . . . . . 12 ∅ = ∅
9190a1i 11 . . . . . . . . . . 11 (𝑦 = ∅ → ∅ = ∅)
9289, 91eqtrd 2655 . . . . . . . . . 10 (𝑦 = ∅ → 𝑦 = ∅)
9392adantl 482 . . . . . . . . 9 ((𝑦 ∈ {∅, {∅}} ∧ 𝑦 = ∅) → 𝑦 = ∅)
94 simpl 473 . . . . . . . . . . 11 ((𝑦 ∈ {∅, {∅}} ∧ ¬ 𝑦 = ∅) → 𝑦 ∈ {∅, {∅}})
9527adantl 482 . . . . . . . . . . 11 ((𝑦 ∈ {∅, {∅}} ∧ ¬ 𝑦 = ∅) → 𝑦 ≠ ∅)
96 elprn1 39297 . . . . . . . . . . 11 ((𝑦 ∈ {∅, {∅}} ∧ 𝑦 ≠ ∅) → 𝑦 = {∅})
9794, 95, 96syl2anc 692 . . . . . . . . . 10 ((𝑦 ∈ {∅, {∅}} ∧ ¬ 𝑦 = ∅) → 𝑦 = {∅})
98 unieq 4415 . . . . . . . . . . 11 (𝑦 = {∅} → 𝑦 = {∅})
991unisn 4422 . . . . . . . . . . . 12 {∅} = ∅
10099a1i 11 . . . . . . . . . . 11 (𝑦 = {∅} → {∅} = ∅)
10198, 100eqtrd 2655 . . . . . . . . . 10 (𝑦 = {∅} → 𝑦 = ∅)
10297, 101syl 17 . . . . . . . . 9 ((𝑦 ∈ {∅, {∅}} ∧ ¬ 𝑦 = ∅) → 𝑦 = ∅)
10393, 102pm2.61dan 831 . . . . . . . 8 (𝑦 ∈ {∅, {∅}} → 𝑦 = ∅)
10488, 103syl 17 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋𝑦) → 𝑦 = ∅)
1052a1i 11 . . . . . . 7 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋𝑦) → ∅ ∈ {∅, 𝑋})
106104, 105eqeltrd 2698 . . . . . 6 (((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋𝑦) → 𝑦 ∈ {∅, 𝑋})
10756, 106pm2.61dan 831 . . . . 5 ((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) → 𝑦 ∈ {∅, 𝑋})
108107a1d 25 . . . 4 ((𝑋𝑉𝑦 ∈ 𝒫 {∅, 𝑋}) → (𝑦 ≼ ω → 𝑦 ∈ {∅, 𝑋}))
109108ralrimiva 2961 . . 3 (𝑋𝑉 → ∀𝑦 ∈ 𝒫 {∅, 𝑋} (𝑦 ≼ ω → 𝑦 ∈ {∅, 𝑋}))
1103, 42, 1093jca 1240 . 2 (𝑋𝑉 → (∅ ∈ {∅, 𝑋} ∧ ∀𝑦 ∈ {∅, 𝑋} ( {∅, 𝑋} ∖ 𝑦) ∈ {∅, 𝑋} ∧ ∀𝑦 ∈ 𝒫 {∅, 𝑋} (𝑦 ≼ ω → 𝑦 ∈ {∅, 𝑋})))
111 prex 4875 . . . 4 {∅, 𝑋} ∈ V
112111a1i 11 . . 3 (𝑋𝑉 → {∅, 𝑋} ∈ V)
113 issal 39867 . . 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3189  cdif 3556  cun 3557  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153  {cpr 4155   cuni 4407   class class class wbr 4618  ωcom 7019  cdom 7905  SAlgcsalg 39861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-pw 4137  df-sn 4154  df-pr 4156  df-uni 4408  df-salg 39862
This theorem is referenced by: (None)
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