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Mirrors > Home > MPE Home > Th. List > preleq | Structured version Visualization version GIF version |
Description: Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |
Ref | Expression |
---|---|
preleq.1 | ⊢ 𝐴 ∈ V |
preleq.2 | ⊢ 𝐵 ∈ V |
preleq.3 | ⊢ 𝐶 ∈ V |
preleq.4 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
preleq | ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preleq.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
2 | preleq.2 | . . . . . . 7 ⊢ 𝐵 ∈ V | |
3 | preleq.3 | . . . . . . 7 ⊢ 𝐶 ∈ V | |
4 | preleq.4 | . . . . . . 7 ⊢ 𝐷 ∈ V | |
5 | 1, 2, 3, 4 | preq12b 4413 | . . . . . 6 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))) |
6 | 5 | biimpi 206 | . . . . 5 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))) |
7 | 6 | ord 391 | . . . 4 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))) |
8 | en2lp 8548 | . . . . 5 ⊢ ¬ (𝐷 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷) | |
9 | eleq12 2720 | . . . . . 6 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 ∈ 𝐵 ↔ 𝐷 ∈ 𝐶)) | |
10 | 9 | anbi1d 741 | . . . . 5 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ↔ (𝐷 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷))) |
11 | 8, 10 | mtbiri 316 | . . . 4 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷)) |
12 | 7, 11 | syl6 35 | . . 3 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷))) |
13 | 12 | con4d 114 | . 2 ⊢ ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
14 | 13 | impcom 445 | 1 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 {cpr 4212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-reg 8538 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-eprel 5058 df-fr 5102 |
This theorem is referenced by: opthreg 8553 dfac2 8991 |
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