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| Description: Alternate proof of iseri 8772, avoiding the usage of mptru 1547 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 8771 | . 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 206 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 Rel wrel 5690 Er wer 8742 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-er 8745 | 
| This theorem is referenced by: (None) | 
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