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Mirrors > Home > MPE Home > Th. List > preleqALT | Structured version Visualization version GIF version |
Description: Alternate proof of preleq 9560, not based on preleqg 9559: Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
preleq.b | ⊢ 𝐵 ∈ V |
preleqALT.d | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
preleqALT | ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preleq.b | . . . . . . . . . 10 ⊢ 𝐵 ∈ V | |
2 | 1 | jctr 526 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V)) |
3 | preleqALT.d | . . . . . . . . . 10 ⊢ 𝐷 ∈ V | |
4 | 3 | jctr 526 | . . . . . . . . 9 ⊢ (𝐶 ∈ 𝐷 → (𝐶 ∈ 𝐷 ∧ 𝐷 ∈ V)) |
5 | preq12bg 4815 | . . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ 𝐷 ∧ 𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))) | |
6 | 2, 4, 5 | syl2an 597 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))) |
7 | 6 | biimpa 478 | . . . . . . 7 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))) |
8 | 7 | ord 863 | . . . . . 6 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))) |
9 | en2lp 9550 | . . . . . . 7 ⊢ ¬ (𝐷 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷) | |
10 | eleq12 2824 | . . . . . . . 8 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → (𝐴 ∈ 𝐵 ↔ 𝐷 ∈ 𝐶)) | |
11 | 10 | anbi1d 631 | . . . . . . 7 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ↔ (𝐷 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷))) |
12 | 9, 11 | mtbiri 327 | . . . . . 6 ⊢ ((𝐴 = 𝐷 ∧ 𝐵 = 𝐶) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷)) |
13 | 8, 12 | syl6 35 | . . . . 5 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷))) |
14 | 13 | con4d 115 | . . . 4 ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
15 | 14 | ex 414 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))) |
16 | 15 | pm2.43a 54 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
17 | 16 | imp 408 | 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 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 Vcvv 3447 {cpr 4592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-reg 9536 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-eprel 5541 df-fr 5592 |
This theorem is referenced by: (None) |
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