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Mirrors > Home > MPE Home > Th. List > Mathboxes > sprid | Structured version Visualization version GIF version |
Description: Two identical representations of the class of all unordered pairs. (Contributed by AV, 21-Nov-2021.) |
Ref | Expression |
---|---|
sprid | ⊢ {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexv 3518 | . . 3 ⊢ (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}) | |
2 | rexv 3518 | . . . 4 ⊢ (∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏 𝑝 = {𝑎, 𝑏}) | |
3 | 2 | exbii 1839 | . . 3 ⊢ (∃𝑎∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) |
4 | 1, 3 | bitri 276 | . 2 ⊢ (∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}) |
5 | 4 | abbii 2883 | 1 ⊢ {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∃wex 1771 {cab 2796 ∃wrex 3136 Vcvv 3492 {cpr 4559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-rex 3141 df-v 3494 |
This theorem is referenced by: (None) |
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