Theorem carden2 9911

Description: Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 10473, 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 9901	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
2		carddom2 9901	. . . 4 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ 𝐵 ≼ 𝐴))
3	2	ancoms 458	. . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ 𝐵 ≼ 𝐴))
4	1, 3	anbi12d 633	. 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐵) ⊆ (card‘𝐴)) ↔ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴)))
5		eqss 3951	. . 3 ⊢ ((card‘𝐴) = (card‘𝐵) ↔ ((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐵) ⊆ (card‘𝐴)))
6	5	bicomi 224	. 2 ⊢ (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐵) ⊆ (card‘𝐴)) ↔ (card‘𝐴) = (card‘𝐵))
7		sbthb 9038	. 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) ↔ 𝐴 ≈ 𝐵)
8	4, 6, 7	3bitr3g 313	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴 ≈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903   class class class wbr 5100  dom cdm 5632  ‘cfv 6500   ≈ cen 8892   ≼ cdom 8893  cardccrd 9859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-card 9863

This theorem is referenced by:  cardsdom2  9912  pm54.43lem  9924  sdom2en01  10224  fin23lem22  10249  fin1a2lem9  10330  pwfseqlem4  10585  hashen  14282