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Mirrors > Home > ILE Home > Th. List > phplem3 | GIF version |
Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. For a version without the redundant hypotheses, see phplem3g 6849. (Contributed by NM, 26-May-1998.) |
Ref | Expression |
---|---|
phplem2.1 | ⊢ 𝐴 ∈ V |
phplem2.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
phplem3 | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsuci 4399 | . 2 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) | |
2 | phplem2.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | phplem2.2 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | phplem2 6846 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
5 | 2 | enref 6758 | . . . 4 ⊢ 𝐴 ≈ 𝐴 |
6 | nnord 4607 | . . . . . 6 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
7 | orddif 4542 | . . . . . 6 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) | |
8 | 6, 7 | syl 14 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
9 | sneq 3602 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
10 | 9 | difeq2d 3253 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵})) |
11 | 10 | eqcoms 2180 | . . . . 5 ⊢ (𝐵 = 𝐴 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵})) |
12 | 8, 11 | sylan9eq 2230 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 = (suc 𝐴 ∖ {𝐵})) |
13 | 5, 12 | breqtrid 4037 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
14 | 4, 13 | jaodan 797 | . 2 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
15 | 1, 14 | sylan2 286 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 708 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∖ cdif 3126 {csn 3591 class class class wbr 4000 Ord word 4358 suc csuc 4361 ωcom 4585 ≈ cen 6731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-tr 4099 df-id 4289 df-iord 4362 df-on 4364 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-en 6734 |
This theorem is referenced by: phplem4 6848 phplem3g 6849 |
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