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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 8653 (via the preleq 8675 step). See df-op 4323 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 4433 | . . . 4 ⊢ 𝐴 ∈ {𝐴, 𝐵} |
3 | opthreg.3 | . . . . 5 ⊢ 𝐶 ∈ V | |
4 | 3 | prid1 4433 | . . . 4 ⊢ 𝐶 ∈ {𝐶, 𝐷} |
5 | prex 5037 | . . . . 5 ⊢ {𝐴, 𝐵} ∈ V | |
6 | 5 | preleq 8675 | . . . 4 ⊢ (((𝐴 ∈ {𝐴, 𝐵} ∧ 𝐶 ∈ {𝐶, 𝐷}) ∧ {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷})) |
7 | 2, 4, 6 | mpanl12 682 | . . 3 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} → (𝐴 = 𝐶 ∧ {𝐴, 𝐵} = {𝐶, 𝐷})) |
8 | preq1 4404 | . . . . . 6 ⊢ (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵}) | |
9 | 8 | eqeq1d 2773 | . . . . 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 4512 | . . . . 5 ⊢ ({𝐶, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷) |
13 | 9, 12 | syl6bi 243 | . . . 4 ⊢ (𝐴 = 𝐶 → ({𝐴, 𝐵} = {𝐶, 𝐷} → 𝐵 = 𝐷)) |
14 | 13 | imdistani 558 | . . 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 4404 | . . . 4 ⊢ (𝐴 = 𝐶 → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}}) | |
17 | 16 | adantr 466 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐴, 𝐵}}) |
18 | preq12 4406 | . . . 4 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷}) | |
19 | 18 | preq2d 4411 | . . 3 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐶, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) |
20 | 17, 19 | eqtrd 2805 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → {𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}}) |
21 | 15, 20 | impbii 199 | 1 ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 {cpr 4318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 ax-reg 8653 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-br 4787 df-opab 4847 df-eprel 5162 df-fr 5208 |
This theorem is referenced by: (None) |
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