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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 4795 | . . . . . 6 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐵〉 = 〈𝐵, 𝐵〉) | |
3 | 2 | sneqd 4569 | . . . . 5 ⊢ (𝐴 = 𝐵 → {〈𝐴, 𝐵〉} = {〈𝐵, 𝐵〉}) |
4 | funsndifnop.b | . . . . . 6 ⊢ 𝐵 ∈ V | |
5 | 4 | snopeqopsnid 5390 | . . . . 5 ⊢ {〈𝐵, 𝐵〉} = 〈{𝐵}, {𝐵}〉 |
6 | 3, 5 | syl6eq 2869 | . . . 4 ⊢ (𝐴 = 𝐵 → {〈𝐴, 𝐵〉} = 〈{𝐵}, {𝐵}〉) |
7 | 1, 6 | syl5eq 2865 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐺 = 〈{𝐵}, {𝐵}〉) |
8 | snex 5322 | . . . 4 ⊢ {𝐵} ∈ V | |
9 | 8, 8 | opelvv 5587 | . . 3 ⊢ 〈{𝐵}, {𝐵}〉 ∈ (V × V) |
10 | 7, 9 | syl6eqel 2918 | . 2 ⊢ (𝐴 = 𝐵 → 𝐺 ∈ (V × V)) |
11 | funsndifnop.a | . . . 4 ⊢ 𝐴 ∈ V | |
12 | 11, 4, 1 | funsndifnop 6905 | . . 3 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐺 ∈ (V × V)) |
13 | 12 | necon4ai 3044 | . 2 ⊢ (𝐺 ∈ (V × V) → 𝐴 = 𝐵) |
14 | 10, 13 | impbii 210 | 1 ⊢ (𝐴 = 𝐵 ↔ 𝐺 ∈ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 〈cop 4563 × cxp 5546 |
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-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 |
This theorem is referenced by: (None) |
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