![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > phpeqd | Structured version Visualization version GIF version |
Description: Corollary of the Pigeonhole Principle using equality. Strengthening of php 9209 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) Avoid ax-pow. (Revised by BTernaryTau, 28-Nov-2024.) |
Ref | Expression |
---|---|
phpeqd.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
phpeqd.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
phpeqd.3 | ⊢ (𝜑 → 𝐴 ≈ 𝐵) |
Ref | Expression |
---|---|
phpeqd | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phpeqd.3 | . 2 ⊢ (𝜑 → 𝐴 ≈ 𝐵) | |
2 | phpeqd.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
3 | phpeqd.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
4 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ⊆ 𝐴) |
5 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵) | |
6 | 5 | neqcomd 2736 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐵 = 𝐴) |
7 | dfpss2 4080 | . . . . . 6 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐵 = 𝐴)) | |
8 | 4, 6, 7 | sylanbrc 582 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ⊊ 𝐴) |
9 | php3 9211 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) | |
10 | 2, 8, 9 | syl2an2r 682 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ≺ 𝐴) |
11 | sdomnen 8976 | . . . . 5 ⊢ (𝐵 ≺ 𝐴 → ¬ 𝐵 ≈ 𝐴) | |
12 | ensymfib 9186 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)) | |
13 | 12 | notbid 318 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (¬ 𝐴 ≈ 𝐵 ↔ ¬ 𝐵 ≈ 𝐴)) |
14 | 13 | biimpar 477 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ≈ 𝐴) → ¬ 𝐴 ≈ 𝐵) |
15 | 2, 11, 14 | syl2an 595 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≺ 𝐴) → ¬ 𝐴 ≈ 𝐵) |
16 | 10, 15 | syldan 590 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 ≈ 𝐵) |
17 | 16 | ex 412 | . 2 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → ¬ 𝐴 ≈ 𝐵)) |
18 | 1, 17 | mt4d 117 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 ⊊ wpss 3944 class class class wbr 5141 ≈ cen 8935 ≺ csdm 8937 Fincfn 8938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 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-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-om 7852 df-1o 8464 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 |
This theorem is referenced by: phphashd 14431 simpgnsgd 20020 |
Copyright terms: Public domain | W3C validator |