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| Mirrors > Home > MPE Home > Th. List > Mathboxes > hfext | Structured version Visualization version GIF version | ||
| Description: Extensionality for HF sets depends only on comparison of HF elements. (Contributed by Scott Fenton, 16-Jul-2015.) |
| Ref | Expression |
|---|---|
| hfext | ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ Hf (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfcleq 2730 | . . 3 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
| 2 | unvdif 4475 | . . . . 5 ⊢ ( Hf ∪ (V ∖ Hf )) = V | |
| 3 | 2 | raleqi 3324 | . . . 4 ⊢ (∀𝑥 ∈ ( Hf ∪ (V ∖ Hf ))(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ V (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
| 4 | ralv 3508 | . . . 4 ⊢ (∀𝑥 ∈ V (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
| 5 | 3, 4 | bitr2i 276 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ ( Hf ∪ (V ∖ Hf ))(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
| 6 | ralunb 4197 | . . 3 ⊢ (∀𝑥 ∈ ( Hf ∪ (V ∖ Hf ))(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ (∀𝑥 ∈ Hf (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ ∀𝑥 ∈ (V ∖ Hf )(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) | |
| 7 | 1, 5, 6 | 3bitri 297 | . 2 ⊢ (𝐴 = 𝐵 ↔ (∀𝑥 ∈ Hf (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ ∀𝑥 ∈ (V ∖ Hf )(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) |
| 8 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 9 | eldif 3961 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ Hf ) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ Hf )) | |
| 10 | 8, 9 | mpbiran 709 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ Hf ) ↔ ¬ 𝑥 ∈ Hf ) |
| 11 | hfelhf 36182 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ Hf ) → 𝑥 ∈ Hf ) | |
| 12 | 11 | stoic1b 1773 | . . . . . . 7 ⊢ ((𝐴 ∈ Hf ∧ ¬ 𝑥 ∈ Hf ) → ¬ 𝑥 ∈ 𝐴) |
| 13 | 12 | adantlr 715 | . . . . . 6 ⊢ (((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) ∧ ¬ 𝑥 ∈ Hf ) → ¬ 𝑥 ∈ 𝐴) |
| 14 | hfelhf 36182 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ Hf ) → 𝑥 ∈ Hf ) | |
| 15 | 14 | stoic1b 1773 | . . . . . . 7 ⊢ ((𝐵 ∈ Hf ∧ ¬ 𝑥 ∈ Hf ) → ¬ 𝑥 ∈ 𝐵) |
| 16 | 15 | adantll 714 | . . . . . 6 ⊢ (((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) ∧ ¬ 𝑥 ∈ Hf ) → ¬ 𝑥 ∈ 𝐵) |
| 17 | 13, 16 | 2falsed 376 | . . . . 5 ⊢ (((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) ∧ ¬ 𝑥 ∈ Hf ) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
| 18 | 10, 17 | sylan2b 594 | . . . 4 ⊢ (((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) ∧ 𝑥 ∈ (V ∖ Hf )) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
| 19 | 18 | ralrimiva 3146 | . . 3 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → ∀𝑥 ∈ (V ∖ Hf )(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
| 20 | 19 | biantrud 531 | . 2 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (∀𝑥 ∈ Hf (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ (∀𝑥 ∈ Hf (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ ∀𝑥 ∈ (V ∖ Hf )(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))) |
| 21 | 7, 20 | bitr4id 290 | 1 ⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ Hf (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 Hf chf 36173 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-reg 9632 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-r1 9804 df-rank 9805 df-hf 36174 |
| This theorem is referenced by: (None) |
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