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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 9529 (via the preleq 9553 step). See df-op 4594 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) (Proof shortened by AV, 15-Jun-2022.) |
Ref | Expression |
---|---|
opthreg.1 | ⊢ 𝐴 ∈ V |
opthreg.2 | ⊢ 𝐵 ∈ V |
opthreg.3 | ⊢ 𝐶 ∈ V |
opthreg.4 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
opthreg | ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opthreg.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
2 | 1 | prid1 4724 | . . . 4 ⊢ 𝐴 ∈ {𝐴, 𝐵} |
3 | opthreg.3 | . . . . 5 ⊢ 𝐶 ∈ V | |
4 | 3 | prid1 4724 | . . . 4 ⊢ 𝐶 ∈ {𝐶, 𝐷} |
5 | prex 5390 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ V | |
6 | 5 | preleq 9553 | . . . 4 ⊢ (((𝐴 ∈ {𝐴, 𝐵} ∧ 𝐶 ∈ {𝐶, 𝐷}) ∧ {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷})) |
7 | 2, 4, 6 | mpanl12 701 | . . 3 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷})) |
8 | preq1 4695 | . . . . . 6 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵}) | |
9 | 8 | eqeq1d 2739 | . . . . 5 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐶, 𝐵} = {𝐶, 𝐷})) |
10 | opthreg.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
11 | opthreg.4 | . . . . . 6 ⊢ 𝐷 ∈ V | |
12 | 10, 11 | preqr2 4808 | . . . . 5 ⊢ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷) |
13 | 9, 12 | syl6bi 253 | . . . 4 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)) |
14 | 13 | imdistani 570 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
15 | 7, 14 | syl 17 | . 2 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
16 | preq1 4695 | . . . 4 ⊢ (𝐴 = 𝐶 → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}}) | |
17 | 16 | adantr 482 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}}) |
18 | preq12 4697 | . . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) | |
19 | 18 | preq2d 4702 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐶, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) |
20 | 17, 19 | eqtrd 2777 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) |
21 | 15, 20 | impbii 208 | 1 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3446 {cpr 4589 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-reg 9529 |
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 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-eprel 5538 df-fr 5589 |
This theorem is referenced by: (None) |
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