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Mirrors > Home > MPE Home > Th. List > funsneqopb | Structured version Visualization version GIF version |
Description: A singleton of an ordered pair is an ordered pair iff the components are equal. (Contributed by AV, 24-Sep-2020.) (Avoid depending on this detail.) |
Ref | Expression |
---|---|
funsndifnop.a | ⊢ 𝐴 ∈ V |
funsndifnop.b | ⊢ 𝐵 ∈ V |
funsndifnop.g | ⊢ 𝐺 = {〈𝐴, 𝐵〉} |
Ref | Expression |
---|---|
funsneqopb | ⊢ (𝐴 = 𝐵 ↔ 𝐺 ∈ (V × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funsndifnop.g | . . . 4 ⊢ 𝐺 = {〈𝐴, 𝐵〉} | |
2 | opeq1 4878 | . . . . . 6 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐵〉 = 〈𝐵, 𝐵〉) | |
3 | 2 | sneqd 4643 | . . . . 5 ⊢ (𝐴 = 𝐵 → {〈𝐴, 𝐵〉} = {〈𝐵, 𝐵〉}) |
4 | funsndifnop.b | . . . . . 6 ⊢ 𝐵 ∈ V | |
5 | 4 | snopeqopsnid 5519 | . . . . 5 ⊢ {〈𝐵, 𝐵〉} = 〈{𝐵}, {𝐵}〉 |
6 | 3, 5 | eqtrdi 2791 | . . . 4 ⊢ (𝐴 = 𝐵 → {〈𝐴, 𝐵〉} = 〈{𝐵}, {𝐵}〉) |
7 | 1, 6 | eqtrid 2787 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐺 = 〈{𝐵}, {𝐵}〉) |
8 | snex 5442 | . . . 4 ⊢ {𝐵} ∈ V | |
9 | 8, 8 | opelvv 5729 | . . 3 ⊢ 〈{𝐵}, {𝐵}〉 ∈ (V × V) |
10 | 7, 9 | eqeltrdi 2847 | . 2 ⊢ (𝐴 = 𝐵 → 𝐺 ∈ (V × V)) |
11 | funsndifnop.a | . . . 4 ⊢ 𝐴 ∈ V | |
12 | 11, 4, 1 | funsndifnop 7171 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐺 ∈ (V × V)) |
13 | 12 | necon4ai 2970 | . 2 ⊢ (𝐺 ∈ (V × V) → 𝐴 = 𝐵) |
14 | 10, 13 | impbii 209 | 1 ⊢ (𝐴 = 𝐵 ↔ 𝐺 ∈ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 〈cop 4637 × cxp 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 |
This theorem is referenced by: (None) |
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