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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 2725 | . . 3 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
2 | unvdif 4474 | . . . . 5 ⊢ ( Hf ∪ (V ∖ Hf )) = V | |
3 | 2 | raleqi 3323 | . . . 4 ⊢ (∀𝑥 ∈ ( Hf ∪ (V ∖ Hf ))(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ V (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
4 | ralv 3498 | . . . 4 ⊢ (∀𝑥 ∈ V (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
5 | 3, 4 | bitr2i 275 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ ( Hf ∪ (V ∖ Hf ))(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
6 | ralunb 4191 | . . 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 296 | . 2 ⊢ (𝐴 = 𝐵 ↔ (∀𝑥 ∈ Hf (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ∧ ∀𝑥 ∈ (V ∖ Hf )(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) |
8 | vex 3478 | . . . . . 6 ⊢ 𝑥 ∈ V | |
9 | eldif 3958 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ Hf ) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ Hf )) | |
10 | 8, 9 | mpbiran 707 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ Hf ) ↔ ¬ 𝑥 ∈ Hf ) |
11 | hfelhf 35148 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ Hf ) → 𝑥 ∈ Hf ) | |
12 | 11 | stoic1b 1775 | . . . . . . 7 ⊢ ((𝐴 ∈ Hf ∧ ¬ 𝑥 ∈ Hf ) → ¬ 𝑥 ∈ 𝐴) |
13 | 12 | adantlr 713 | . . . . . 6 ⊢ (((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) ∧ ¬ 𝑥 ∈ Hf ) → ¬ 𝑥 ∈ 𝐴) |
14 | hfelhf 35148 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐵 ∈ Hf ) → 𝑥 ∈ Hf ) | |
15 | 14 | stoic1b 1775 | . . . . . . 7 ⊢ ((𝐵 ∈ Hf ∧ ¬ 𝑥 ∈ Hf ) → ¬ 𝑥 ∈ 𝐵) |
16 | 15 | adantll 712 | . . . . . 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 532 | . 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 289 | 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 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ∖ cdif 3945 ∪ cun 3946 Hf chf 35139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-reg 9586 ax-inf2 9635 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-r1 9758 df-rank 9759 df-hf 35140 |
This theorem is referenced by: (None) |
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