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Mirrors > Home > MPE Home > Th. List > phpeqd | Structured version Visualization version GIF version |
Description: Corollary of the Pigeonhole Principle using equality. Strengthening of php 9161 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 482 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ⊆ 𝐴) |
5 | simpr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵) | |
6 | 5 | neqcomd 2747 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐵 = 𝐴) |
7 | dfpss2 4050 | . . . . . 6 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐵 = 𝐴)) | |
8 | 4, 6, 7 | sylanbrc 584 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ⊊ 𝐴) |
9 | php3 9163 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) | |
10 | 2, 8, 9 | syl2an2r 684 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ≺ 𝐴) |
11 | sdomnen 8928 | . . . . 5 ⊢ (𝐵 ≺ 𝐴 → ¬ 𝐵 ≈ 𝐴) | |
12 | ensymfib 9138 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)) | |
13 | 12 | notbid 318 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (¬ 𝐴 ≈ 𝐵 ↔ ¬ 𝐵 ≈ 𝐴)) |
14 | 13 | biimpar 479 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ≈ 𝐴) → ¬ 𝐴 ≈ 𝐵) |
15 | 2, 11, 14 | syl2an 597 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ≺ 𝐴) → ¬ 𝐴 ≈ 𝐵) |
16 | 10, 15 | syldan 592 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 ≈ 𝐵) |
17 | 16 | ex 414 | . 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 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3915 ⊊ wpss 3916 class class class wbr 5110 ≈ cen 8887 ≺ csdm 8889 Fincfn 8890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-om 7808 df-1o 8417 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 |
This theorem is referenced by: phphashd 14372 simpgnsgd 19886 |
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