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Mirrors > Home > MPE Home > Th. List > carden2a | Structured version Visualization version GIF version |
Description: If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 9003 are meant to replace carden 9585 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) |
Ref | Expression |
---|---|
carden2a | ⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2933 | . 2 ⊢ ((card‘𝐴) ≠ ∅ ↔ ¬ (card‘𝐴) = ∅) | |
2 | ndmfv 6380 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅) | |
3 | eqeq1 2764 | . . . . . . 7 ⊢ ((card‘𝐴) = (card‘𝐵) → ((card‘𝐴) = ∅ ↔ (card‘𝐵) = ∅)) | |
4 | 2, 3 | syl5ibr 236 | . . . . . 6 ⊢ ((card‘𝐴) = (card‘𝐵) → (¬ 𝐵 ∈ dom card → (card‘𝐴) = ∅)) |
5 | 4 | con1d 139 | . . . . 5 ⊢ ((card‘𝐴) = (card‘𝐵) → (¬ (card‘𝐴) = ∅ → 𝐵 ∈ dom card)) |
6 | 5 | imp 444 | . . . 4 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐵 ∈ dom card) |
7 | cardid2 8989 | . . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → (card‘𝐵) ≈ 𝐵) |
9 | cardid2 8989 | . . . . . . 7 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
10 | ndmfv 6380 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅) | |
11 | 9, 10 | nsyl4 156 | . . . . . 6 ⊢ (¬ (card‘𝐴) = ∅ → (card‘𝐴) ≈ 𝐴) |
12 | 11 | ensymd 8174 | . . . . 5 ⊢ (¬ (card‘𝐴) = ∅ → 𝐴 ≈ (card‘𝐴)) |
13 | breq2 4808 | . . . . . 6 ⊢ ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) ↔ 𝐴 ≈ (card‘𝐵))) | |
14 | entr 8175 | . . . . . . 7 ⊢ ((𝐴 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≈ 𝐵) | |
15 | 14 | ex 449 | . . . . . 6 ⊢ (𝐴 ≈ (card‘𝐵) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵)) |
16 | 13, 15 | syl6bi 243 | . . . . 5 ⊢ ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵))) |
17 | 12, 16 | syl5 34 | . . . 4 ⊢ ((card‘𝐴) = (card‘𝐵) → (¬ (card‘𝐴) = ∅ → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵))) |
18 | 17 | imp 444 | . . 3 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵)) |
19 | 8, 18 | mpd 15 | . 2 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐴 ≈ 𝐵) |
20 | 1, 19 | sylan2b 493 | 1 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∅c0 4058 class class class wbr 4804 dom cdm 5266 ‘cfv 6049 ≈ cen 8120 cardccrd 8971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-ord 5887 df-on 5888 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-er 7913 df-en 8124 df-card 8975 |
This theorem is referenced by: card1 9004 |
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