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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 5368). (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 34317 | . . . 4 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr1 ⦅𝐴, 𝐵⦆ = pr1 ⦅𝐶, 𝐷⦆) | |
2 | bj-pr21val 34328 | . . . 4 ⊢ pr1 ⦅𝐴, 𝐵⦆ = 𝐴 | |
3 | bj-pr21val 34328 | . . . 4 ⊢ pr1 ⦅𝐶, 𝐷⦆ = 𝐶 | |
4 | 1, 2, 3 | 3eqtr3g 2879 | . . 3 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐴 = 𝐶) |
5 | bj-pr2eq 34331 | . . . 4 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → pr2 ⦅𝐴, 𝐵⦆ = pr2 ⦅𝐶, 𝐷⦆) | |
6 | bj-pr22val 34334 | . . . 4 ⊢ pr2 ⦅𝐴, 𝐵⦆ = 𝐵 | |
7 | bj-pr22val 34334 | . . . 4 ⊢ pr2 ⦅𝐶, 𝐷⦆ = 𝐷 | |
8 | 5, 6, 7 | 3eqtr3g 2879 | . . 3 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → 𝐵 = 𝐷) |
9 | 4, 8 | jca 514 | . 2 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
10 | bj-2upleq 34327 | . . 3 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆)) | |
11 | 10 | imp 409 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → ⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆) |
12 | 9, 11 | impbii 211 | 1 ⊢ (⦅𝐴, 𝐵⦆ = ⦅𝐶, 𝐷⦆ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 pr1 bj-cpr1 34315 ⦅bj-c2uple 34325 pr2 bj-cpr2 34329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-suc 6197 df-1o 8102 df-bj-sngl 34281 df-bj-tag 34290 df-bj-proj 34306 df-bj-1upl 34313 df-bj-pr1 34316 df-bj-2upl 34326 df-bj-pr2 34330 |
This theorem is referenced by: (None) |
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