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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sprssspr | Structured version Visualization version GIF version |
Description: The set of all unordered pairs over a given set 𝑉 is a subset of the set of all unordered pairs. (Contributed by AV, 21-Nov-2021.) |
Ref | Expression |
---|---|
sprssspr | ⊢ (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sprval 45761 | . . 3 ⊢ (𝑉 ∈ V → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) | |
2 | r2ex 3189 | . . . . . . 7 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑝 = {𝑎, 𝑏})) | |
3 | simpr 486 | . . . . . . . 8 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑝 = {𝑎, 𝑏}) → 𝑝 = {𝑎, 𝑏}) | |
4 | 3 | 2eximi 1839 | . . . . . . 7 ⊢ (∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑝 = {𝑎, 𝑏}) → ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) |
5 | 2, 4 | sylbi 216 | . . . . . 6 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} → ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) |
6 | 5 | ax-gen 1798 | . . . . 5 ⊢ ∀𝑝(∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} → ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝑉 ∈ V → ∀𝑝(∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} → ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})) |
8 | ss2ab 4020 | . . . 4 ⊢ ({𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ∀𝑝(∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} → ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})) | |
9 | 7, 8 | sylibr 233 | . . 3 ⊢ (𝑉 ∈ V → {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}) |
10 | 1, 9 | eqsstrd 3986 | . 2 ⊢ (𝑉 ∈ V → (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}) |
11 | fvprc 6838 | . . 3 ⊢ (¬ 𝑉 ∈ V → (Pairs‘𝑉) = ∅) | |
12 | 0ss 4360 | . . . 4 ⊢ ∅ ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} | |
13 | 12 | a1i 11 | . . 3 ⊢ (¬ 𝑉 ∈ V → ∅ ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}) |
14 | 11, 13 | eqsstrd 3986 | . 2 ⊢ (¬ 𝑉 ∈ V → (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}) |
15 | 10, 14 | pm2.61i 182 | 1 ⊢ (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∀wal 1540 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ∃wrex 3070 Vcvv 3447 ⊆ wss 3914 ∅c0 4286 {cpr 4592 ‘cfv 6500 Pairscspr 45759 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-spr 45760 |
This theorem is referenced by: spr0el 45764 |
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