![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iseriALT | Structured version Visualization version GIF version |
Description: Alternate proof of iseri 8729, avoiding the usage of mptru 1540 and ⊤ as antecedent by using ax-mp 5 and one of the hypotheses as antecedent. This results, however, in a slightly longer proof. (Contributed by AV, 30-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
iseri.1 | ⊢ Rel 𝑅 |
iseri.2 | ⊢ (𝑥𝑅𝑦 → 𝑦𝑅𝑥) |
iseri.3 | ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) |
iseri.4 | ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥) |
Ref | Expression |
---|---|
iseriALT | ⊢ 𝑅 Er 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseri.1 | . 2 ⊢ Rel 𝑅 | |
2 | id 22 | . . 3 ⊢ (Rel 𝑅 → Rel 𝑅) | |
3 | iseri.2 | . . . 4 ⊢ (𝑥𝑅𝑦 → 𝑦𝑅𝑥) | |
4 | 3 | adantl 481 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥) |
5 | iseri.3 | . . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) | |
6 | 5 | adantl 481 | . . 3 ⊢ ((Rel 𝑅 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧) |
7 | iseri.4 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥) | |
8 | 7 | a1i 11 | . . 3 ⊢ (Rel 𝑅 → (𝑥 ∈ 𝐴 ↔ 𝑥𝑅𝑥)) |
9 | 2, 4, 6, 8 | iserd 8728 | . 2 ⊢ (Rel 𝑅 → 𝑅 Er 𝐴) |
10 | 1, 9 | ax-mp 5 | 1 ⊢ 𝑅 Er 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2098 class class class wbr 5141 Rel wrel 5674 Er wer 8699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-er 8702 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |