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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 9040 (via the preleq 9063 step). See df-op 4532 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 4658 | . . . 4 ⊢ 𝐴 ∈ {𝐴, 𝐵} |
3 | opthreg.3 | . . . . 5 ⊢ 𝐶 ∈ V | |
4 | 3 | prid1 4658 | . . . 4 ⊢ 𝐶 ∈ {𝐶, 𝐷} |
5 | prex 5298 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ V | |
6 | 5 | preleq 9063 | . . . 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 4629 | . . . . . 6 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵}) | |
9 | 8 | eqeq1d 2800 | . . . . 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 4740 | . . . . 5 ⊢ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷) |
13 | 9, 12 | syl6bi 256 | . . . 4 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)) |
14 | 13 | imdistani 572 | . . 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 4629 | . . . 4 ⊢ (𝐴 = 𝐶 → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}}) | |
17 | 16 | adantr 484 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}}) |
18 | preq12 4631 | . . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) | |
19 | 18 | preq2d 4636 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐶, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) |
20 | 17, 19 | eqtrd 2833 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) |
21 | 15, 20 | impbii 212 | 1 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {cpr 4527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-reg 9040 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-eprel 5430 df-fr 5478 |
This theorem is referenced by: (None) |
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