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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ideqg1ALT | Structured version Visualization version GIF version |
Description: Alternate proof of
bj-ideqg1 using brabga 5414 instead of the "unbounded"
version bj-brab2a1 34463 or brab2a 5637. (Contributed by BJ, 25-Dec-2023.)
(Proof modification is discouraged.) (New usage is discouraged.)
TODO: delete once bj-ideqg 34471 is in the main section. |
Ref | Expression |
---|---|
bj-ideqg1ALT | ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reli 5691 | . . . 4 ⊢ Rel I | |
2 | 1 | brrelex12i 5600 | . . 3 ⊢ (𝐴 I 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | 2 | adantl 484 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
4 | elex 3509 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
5 | 4 | adantr 483 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V) |
6 | eleq1 2899 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊)) | |
7 | 6 | biimparc 482 | . . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑊) |
8 | 7 | elexd 3511 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V) |
9 | 5, 8 | jaoian 953 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 = 𝐵) → 𝐴 ∈ V) |
10 | eleq1 2899 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉)) | |
11 | 10 | biimpac 481 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑉) |
12 | 11 | elexd 3511 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V) |
13 | elex 3509 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V) | |
14 | 13 | adantr 483 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V) |
15 | 12, 14 | jaoian 953 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 = 𝐵) → 𝐵 ∈ V) |
16 | 9, 15 | jca 514 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
17 | eqeq12 2834 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵)) | |
18 | df-id 5453 | . . 3 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
19 | 17, 18 | brabga 5414 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) |
20 | 3, 16, 19 | pm5.21nd 800 | 1 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 Vcvv 3491 class class class wbr 5059 I cid 5452 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 |
This theorem is referenced by: (None) |
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