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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 9396 are meant to replace carden 9973 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 3017 | . 2 ⊢ ((card‘𝐴) ≠ ∅ ↔ ¬ (card‘𝐴) = ∅) | |
2 | ndmfv 6700 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅) | |
3 | eqeq1 2825 | . . . . . . 7 ⊢ ((card‘𝐴) = (card‘𝐵) → ((card‘𝐴) = ∅ ↔ (card‘𝐵) = ∅)) | |
4 | 2, 3 | syl5ibr 248 | . . . . . 6 ⊢ ((card‘𝐴) = (card‘𝐵) → (¬ 𝐵 ∈ dom card → (card‘𝐴) = ∅)) |
5 | 4 | con1d 147 | . . . . 5 ⊢ ((card‘𝐴) = (card‘𝐵) → (¬ (card‘𝐴) = ∅ → 𝐵 ∈ dom card)) |
6 | 5 | imp 409 | . . . 4 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐵 ∈ dom card) |
7 | cardid2 9382 | . . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → (card‘𝐵) ≈ 𝐵) |
9 | breq2 5070 | . . . . 5 ⊢ ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) ↔ 𝐴 ≈ (card‘𝐵))) | |
10 | entr 8561 | . . . . . 6 ⊢ ((𝐴 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≈ 𝐵) | |
11 | 10 | ex 415 | . . . . 5 ⊢ (𝐴 ≈ (card‘𝐵) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵)) |
12 | 9, 11 | syl6bi 255 | . . . 4 ⊢ ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵))) |
13 | cardid2 9382 | . . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
14 | ndmfv 6700 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅) | |
15 | 13, 14 | nsyl4 161 | . . . . 5 ⊢ (¬ (card‘𝐴) = ∅ → (card‘𝐴) ≈ 𝐴) |
16 | 15 | ensymd 8560 | . . . 4 ⊢ (¬ (card‘𝐴) = ∅ → 𝐴 ≈ (card‘𝐴)) |
17 | 12, 16 | impel 508 | . . 3 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → ((card‘𝐵) ≈ 𝐵 → 𝐴 ≈ 𝐵)) |
18 | 8, 17 | mpd 15 | . 2 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐴 ≈ 𝐵) |
19 | 1, 18 | sylan2b 595 | 1 ⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 class class class wbr 5066 dom cdm 5555 ‘cfv 6355 ≈ cen 8506 cardccrd 9364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-er 8289 df-en 8510 df-card 9368 |
This theorem is referenced by: card1 9397 |
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