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Mirrors > Home > MPE Home > Th. List > iseriALT | Structured version Visualization version GIF version |
Description: Alternate proof of iseri 7656, avoiding the usage of trud 1484 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 7655 | . 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 195 ∧ wa 383 ∈ wcel 1977 class class class wbr 4583 Rel wrel 5043 Er wer 7626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-er 7629 |
This theorem is referenced by: (None) |
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