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Mirrors > Home > MPE Home > Th. List > r1pwALT | Structured version Visualization version GIF version |
Description: Alternate shorter proof of r1pw 9822 based on the additional axioms ax-reg 9569 and ax-inf2 9618. (Contributed by Raph Levien, 29-May-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
r1pwALT | ⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2820 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (𝑅1‘𝐵) ↔ 𝐴 ∈ (𝑅1‘𝐵))) | |
2 | pweq 4610 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
3 | 2 | eleq1d 2817 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ (𝑅1‘suc 𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))) |
4 | 1, 3 | bibi12d 345 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝑥 ∈ (𝑅1‘suc 𝐵)) ↔ (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))) |
5 | 4 | imbi2d 340 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ On → (𝑥 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝑥 ∈ (𝑅1‘suc 𝐵))) ↔ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))))) |
6 | vex 3477 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
7 | 6 | rankr1a 9813 | . . . . . 6 ⊢ (𝐵 ∈ On → (𝑥 ∈ (𝑅1‘𝐵) ↔ (rank‘𝑥) ∈ 𝐵)) |
8 | eloni 6363 | . . . . . . 7 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
9 | ordsucelsuc 7793 | . . . . . . 7 ⊢ (Ord 𝐵 → ((rank‘𝑥) ∈ 𝐵 ↔ suc (rank‘𝑥) ∈ suc 𝐵)) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (𝐵 ∈ On → ((rank‘𝑥) ∈ 𝐵 ↔ suc (rank‘𝑥) ∈ suc 𝐵)) |
11 | 7, 10 | bitrd 278 | . . . . 5 ⊢ (𝐵 ∈ On → (𝑥 ∈ (𝑅1‘𝐵) ↔ suc (rank‘𝑥) ∈ suc 𝐵)) |
12 | 6 | rankpw 9820 | . . . . . 6 ⊢ (rank‘𝒫 𝑥) = suc (rank‘𝑥) |
13 | 12 | eleq1i 2823 | . . . . 5 ⊢ ((rank‘𝒫 𝑥) ∈ suc 𝐵 ↔ suc (rank‘𝑥) ∈ suc 𝐵) |
14 | 11, 13 | bitr4di 288 | . . . 4 ⊢ (𝐵 ∈ On → (𝑥 ∈ (𝑅1‘𝐵) ↔ (rank‘𝒫 𝑥) ∈ suc 𝐵)) |
15 | onsuc 7782 | . . . . 5 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On) | |
16 | 6 | pwex 5371 | . . . . . 6 ⊢ 𝒫 𝑥 ∈ V |
17 | 16 | rankr1a 9813 | . . . . 5 ⊢ (suc 𝐵 ∈ On → (𝒫 𝑥 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝑥) ∈ suc 𝐵)) |
18 | 15, 17 | syl 17 | . . . 4 ⊢ (𝐵 ∈ On → (𝒫 𝑥 ∈ (𝑅1‘suc 𝐵) ↔ (rank‘𝒫 𝑥) ∈ suc 𝐵)) |
19 | 14, 18 | bitr4d 281 | . . 3 ⊢ (𝐵 ∈ On → (𝑥 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝑥 ∈ (𝑅1‘suc 𝐵))) |
20 | 5, 19 | vtoclg 3553 | . 2 ⊢ (𝐴 ∈ V → (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))) |
21 | elex 3491 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ V) | |
22 | elex 3491 | . . . . 5 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝒫 𝐴 ∈ V) | |
23 | pwexb 7736 | . . . . 5 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) | |
24 | 22, 23 | sylibr 233 | . . . 4 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc 𝐵) → 𝐴 ∈ V) |
25 | 21, 24 | pm5.21ni 378 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))) |
26 | 25 | a1d 25 | . 2 ⊢ (¬ 𝐴 ∈ V → (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))) |
27 | 20, 26 | pm2.61i 182 | 1 ⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 Vcvv 3473 𝒫 cpw 4596 Ord word 6352 Oncon0 6353 suc csuc 6355 ‘cfv 6532 𝑅1cr1 9739 rankcrnk 9740 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-reg 9569 ax-inf2 9618 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-r1 9741 df-rank 9742 |
This theorem is referenced by: (None) |
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