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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-2uplth | Structured version Visualization version GIF version |
Description: The characteristic property of couples. Note that this holds without sethood hypotheses (compare opth 5333). (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-2uplth | ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-pr1eq 34438 | . . . 4 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr1 ⦅𝐴, 𝐵⦆ = pr1 ⦅𝐶, 𝐷⦆) | |
2 | bj-pr21val 34449 | . . . 4 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴 | |
3 | bj-pr21val 34449 | . . . 4 ⊢ pr1 ⦅𝐶, 𝐷⦆ = 𝐶 | |
4 | 1, 2, 3 | 3eqtr3g 2856 | . . 3 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐴 = 𝐶) |
5 | bj-pr2eq 34452 | . . . 4 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr2 ⦅𝐴, 𝐵⦆ = pr2 ⦅𝐶, 𝐷⦆) | |
6 | bj-pr22val 34455 | . . . 4 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵 | |
7 | bj-pr22val 34455 | . . . 4 ⊢ pr2 ⦅𝐶, 𝐷⦆ = 𝐷 | |
8 | 5, 6, 7 | 3eqtr3g 2856 | . . 3 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐵 = 𝐷) |
9 | 4, 8 | jca 515 | . 2 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
10 | bj-2upleq 34448 | . . 3 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆)) | |
11 | 10 | imp 410 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆) |
12 | 9, 11 | impbii 212 | 1 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 pr1 bj-cpr1 34436 ⦅bj-c2uple 34446 pr2 bj-cpr2 34450 |
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-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 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-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-suc 6165 df-1o 8085 df-bj-sngl 34402 df-bj-tag 34411 df-bj-proj 34427 df-bj-1upl 34434 df-bj-pr1 34437 df-bj-2upl 34447 df-bj-pr2 34451 |
This theorem is referenced by: (None) |
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