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Mirrors > Home > ILE Home > Th. List > opthreg | GIF version |
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4536 (via the preleq 4554 step). See df-op 3601 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (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 |
---|---|
opthreg | ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preleq.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
2 | 1 | prid1 3698 | . . . 4 ⊢ 𝐴 ∈ {𝐴, 𝐵} |
3 | preleq.3 | . . . . 5 ⊢ 𝐶 ∈ V | |
4 | 3 | prid1 3698 | . . . 4 ⊢ 𝐶 ∈ {𝐶, 𝐷} |
5 | preleq.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
6 | prexg 4211 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V) | |
7 | 1, 5, 6 | mp2an 426 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ V |
8 | preleq.4 | . . . . . 6 ⊢ 𝐷 ∈ V | |
9 | prexg 4211 | . . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → {𝐶, 𝐷} ∈ V) | |
10 | 3, 8, 9 | mp2an 426 | . . . . 5 ⊢ {𝐶, 𝐷} ∈ V |
11 | 1, 7, 3, 10 | preleq 4554 | . . . 4 ⊢ (((𝐴 ∈ {𝐴, 𝐵} ∧ 𝐶 ∈ {𝐶, 𝐷}) ∧ {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷})) |
12 | 2, 4, 11 | mpanl12 436 | . . 3 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷})) |
13 | preq1 3669 | . . . . . 6 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵}) | |
14 | 13 | eqeq1d 2186 | . . . . 5 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐶, 𝐵} = {𝐶, 𝐷})) |
15 | 5, 8 | preqr2 3769 | . . . . 5 ⊢ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷) |
16 | 14, 15 | syl6bi 163 | . . . 4 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)) |
17 | 16 | imdistani 445 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
18 | 12, 17 | syl 14 | . 2 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
19 | preq1 3669 | . . . 4 ⊢ (𝐴 = 𝐶 → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}}) | |
20 | 19 | adantr 276 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}}) |
21 | preq12 3671 | . . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) | |
22 | 21 | preq2d 3676 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐶, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) |
23 | 20, 22 | eqtrd 2210 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) |
24 | 18, 23 | impbii 126 | 1 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 Vcvv 2737 {cpr 3593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pr 4209 ax-setind 4536 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-dif 3131 df-un 3133 df-sn 3598 df-pr 3599 |
This theorem is referenced by: (None) |
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