Theorem carden2 9415

Description: Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 9972, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.)

Assertion

Ref	Expression
carden2	⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴 ≈ 𝐵))

Proof of Theorem carden2

Step	Hyp	Ref	Expression
1		carddom2 9405	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
2		carddom2 9405	. . . 4 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ 𝐵 ≼ 𝐴))
3	2	ancoms 461	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ 𝐵 ≼ 𝐴))
4	1, 3	anbi12d 632	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐵) ⊆ (card‘𝐴)) ↔ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)))
5		eqss 3981	. . 3 ⊢ ((card‘𝐴) = (card‘𝐵) ↔ ((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐵) ⊆ (card‘𝐴)))
6	5	bicomi 226	. 2 ⊢ (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐵) ⊆ (card‘𝐴)) ↔ (card‘𝐴) = (card‘𝐵))
7		sbthb 8637	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ 𝐴 ≈ 𝐵)
8	4, 6, 7	3bitr3g 315	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴 ≈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ⊆ wss 3935   class class class wbr 5065  dom cdm 5554  ‘cfv 6354   ≈ cen 8505   ≼ cdom 8506  cardccrd 9363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  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 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-card 9367

This theorem is referenced by:  cardsdom2  9416  pm54.43lem  9427  sdom2en01  9723  fin23lem22  9748  fin1a2lem9  9829  pwfseqlem4  10083  hashen  13706