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Mirrors > Home > ILE Home > Th. List > phplem3g | GIF version |
Description: A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6748 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.) |
Ref | Expression |
---|---|
phplem3g | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2202 | . . . . 5 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ suc 𝐴 ↔ 𝐵 ∈ suc 𝐴)) | |
2 | 1 | anbi2d 459 | . . . 4 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ ω ∧ 𝑏 ∈ suc 𝐴) ↔ (𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴))) |
3 | sneq 3538 | . . . . . 6 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵}) | |
4 | 3 | difeq2d 3194 | . . . . 5 ⊢ (𝑏 = 𝐵 → (suc 𝐴 ∖ {𝑏}) = (suc 𝐴 ∖ {𝐵})) |
5 | 4 | breq2d 3941 | . . . 4 ⊢ (𝑏 = 𝐵 → (𝐴 ≈ (suc 𝐴 ∖ {𝑏}) ↔ 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))) |
6 | 2, 5 | imbi12d 233 | . . 3 ⊢ (𝑏 = 𝐵 → (((𝐴 ∈ ω ∧ 𝑏 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝑏})) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})))) |
7 | eleq1 2202 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ω ↔ 𝐴 ∈ ω)) | |
8 | suceq 4324 | . . . . . . . 8 ⊢ (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴) | |
9 | 8 | eleq2d 2209 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑏 ∈ suc 𝑎 ↔ 𝑏 ∈ suc 𝐴)) |
10 | 7, 9 | anbi12d 464 | . . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ ω ∧ 𝑏 ∈ suc 𝑎) ↔ (𝐴 ∈ ω ∧ 𝑏 ∈ suc 𝐴))) |
11 | id 19 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴) | |
12 | 8 | difeq1d 3193 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → (suc 𝑎 ∖ {𝑏}) = (suc 𝐴 ∖ {𝑏})) |
13 | 11, 12 | breq12d 3942 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ≈ (suc 𝑎 ∖ {𝑏}) ↔ 𝐴 ≈ (suc 𝐴 ∖ {𝑏}))) |
14 | 10, 13 | imbi12d 233 | . . . . 5 ⊢ (𝑎 = 𝐴 → (((𝑎 ∈ ω ∧ 𝑏 ∈ suc 𝑎) → 𝑎 ≈ (suc 𝑎 ∖ {𝑏})) ↔ ((𝐴 ∈ ω ∧ 𝑏 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝑏})))) |
15 | vex 2689 | . . . . . 6 ⊢ 𝑎 ∈ V | |
16 | vex 2689 | . . . . . 6 ⊢ 𝑏 ∈ V | |
17 | 15, 16 | phplem3 6748 | . . . . 5 ⊢ ((𝑎 ∈ ω ∧ 𝑏 ∈ suc 𝑎) → 𝑎 ≈ (suc 𝑎 ∖ {𝑏})) |
18 | 14, 17 | vtoclg 2746 | . . . 4 ⊢ (𝐴 ∈ ω → ((𝐴 ∈ ω ∧ 𝑏 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝑏}))) |
19 | 18 | anabsi5 568 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝑏 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝑏})) |
20 | 6, 19 | vtoclg 2746 | . 2 ⊢ (𝐵 ∈ suc 𝐴 → ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))) |
21 | 20 | anabsi7 570 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∖ cdif 3068 {csn 3527 class class class wbr 3929 suc csuc 4287 ωcom 4504 ≈ cen 6632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-en 6635 |
This theorem is referenced by: phplem4dom 6756 phpm 6759 phplem4on 6761 |
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