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Mirrors > Home > MPE Home > Th. List > opthreg | Structured version Visualization version GIF version |
Description: Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 8497 (via the preleq 8514 step). See df-op 4184 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 4297 | . . . 4 ⊢ 𝐴 ∈ {𝐴, 𝐵} |
3 | preleq.3 | . . . . 5 ⊢ 𝐶 ∈ V | |
4 | 3 | prid1 4297 | . . . 4 ⊢ 𝐶 ∈ {𝐶, 𝐷} |
5 | prex 4909 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ V | |
6 | prex 4909 | . . . . 5 ⊢ {𝐶, 𝐷} ∈ V | |
7 | 1, 5, 3, 6 | preleq 8514 | . . . 4 ⊢ (((𝐴 ∈ {𝐴, 𝐵} ∧ 𝐶 ∈ {𝐶, 𝐷}) ∧ {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷})) |
8 | 2, 4, 7 | mpanl12 718 | . . 3 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷})) |
9 | preq1 4268 | . . . . . 6 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵}) | |
10 | 9 | eqeq1d 2624 | . . . . 5 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐶, 𝐵} = {𝐶, 𝐷})) |
11 | preleq.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
12 | preleq.4 | . . . . . 6 ⊢ 𝐷 ∈ V | |
13 | 11, 12 | preqr2 4381 | . . . . 5 ⊢ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷) |
14 | 10, 13 | syl6bi 243 | . . . 4 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)) |
15 | 14 | imdistani 726 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
16 | 8, 15 | syl 17 | . 2 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
17 | preq1 4268 | . . . 4 ⊢ (𝐴 = 𝐶 → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}}) | |
18 | 17 | adantr 481 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}}) |
19 | preq12 4270 | . . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) | |
20 | 19 | preq2d 4275 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐶, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) |
21 | 18, 20 | eqtrd 2656 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) |
22 | 16, 21 | impbii 199 | 1 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {cpr 4179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-reg 8497 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-eprel 5029 df-fr 5073 |
This theorem is referenced by: (None) |
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