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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 4784 | . . . . . 6 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐵〉 = 〈𝐵, 𝐵〉) | |
3 | 2 | sneqd 4553 | . . . . 5 ⊢ (𝐴 = 𝐵 → {〈𝐴, 𝐵〉} = {〈𝐵, 𝐵〉}) |
4 | funsndifnop.b | . . . . . 6 ⊢ 𝐵 ∈ V | |
5 | 4 | snopeqopsnid 5392 | . . . . 5 ⊢ {〈𝐵, 𝐵〉} = 〈{𝐵}, {𝐵}〉 |
6 | 3, 5 | eqtrdi 2794 | . . . 4 ⊢ (𝐴 = 𝐵 → {〈𝐴, 𝐵〉} = 〈{𝐵}, {𝐵}〉) |
7 | 1, 6 | eqtrid 2789 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐺 = 〈{𝐵}, {𝐵}〉) |
8 | snex 5324 | . . . 4 ⊢ {𝐵} ∈ V | |
9 | 8, 8 | opelvv 5590 | . . 3 ⊢ 〈{𝐵}, {𝐵}〉 ∈ (V × V) |
10 | 7, 9 | eqeltrdi 2846 | . 2 ⊢ (𝐴 = 𝐵 → 𝐺 ∈ (V × V)) |
11 | funsndifnop.a | . . . 4 ⊢ 𝐴 ∈ V | |
12 | 11, 4, 1 | funsndifnop 6966 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐺 ∈ (V × V)) |
13 | 12 | necon4ai 2972 | . 2 ⊢ (𝐺 ∈ (V × V) → 𝐴 = 𝐵) |
14 | 10, 13 | impbii 212 | 1 ⊢ (𝐴 = 𝐵 ↔ 𝐺 ∈ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2110 Vcvv 3408 {csn 4541 〈cop 4547 × cxp 5549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 |
This theorem is referenced by: (None) |
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