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Theorem karden 9932
Description: If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 10588). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 9931 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from {𝑥𝑥𝐴}. (Contributed by NM, 18-Dec-2003.) (Revised by AV, 12-Jul-2022.)
Hypotheses
Ref Expression
karden.a 𝐴 ∈ V
karden.c 𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
karden.d 𝐷 = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
Assertion
Ref Expression
karden (𝐶 = 𝐷𝐴𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem karden
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 karden.a . . . . . . 7 𝐴 ∈ V
2 breq1 5150 . . . . . . 7 (𝑤 = 𝐴 → (𝑤𝐴𝐴𝐴))
31enref 9023 . . . . . . 7 𝐴𝐴
41, 2, 3ceqsexv2d 3532 . . . . . 6 𝑤 𝑤𝐴
5 abn0 4390 . . . . . 6 ({𝑤𝑤𝐴} ≠ ∅ ↔ ∃𝑤 𝑤𝐴)
64, 5mpbir 231 . . . . 5 {𝑤𝑤𝐴} ≠ ∅
7 scott0 9923 . . . . . 6 ({𝑤𝑤𝐴} = ∅ ↔ {𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} = ∅)
87necon3bii 2990 . . . . 5 ({𝑤𝑤𝐴} ≠ ∅ ↔ {𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅)
96, 8mpbi 230 . . . 4 {𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅
10 rabn0 4394 . . . 4 ({𝑧 ∈ {𝑤𝑤𝐴} ∣ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)} ≠ ∅ ↔ ∃𝑧 ∈ {𝑤𝑤𝐴}∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦))
119, 10mpbi 230 . . 3 𝑧 ∈ {𝑤𝑤𝐴}∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)
12 vex 3481 . . . . . . . 8 𝑧 ∈ V
13 breq1 5150 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤𝐴𝑧𝐴))
1412, 13elab 3680 . . . . . . 7 (𝑧 ∈ {𝑤𝑤𝐴} ↔ 𝑧𝐴)
15 breq1 5150 . . . . . . . 8 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
1615ralab 3699 . . . . . . 7 (∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))
1714, 16anbi12i 628 . . . . . 6 ((𝑧 ∈ {𝑤𝑤𝐴} ∧ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) ↔ (𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
18 simpl 482 . . . . . . . . 9 ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐴)
1918a1i 11 . . . . . . . 8 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐴))
20 karden.c . . . . . . . . . . . 12 𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
21 karden.d . . . . . . . . . . . 12 𝐷 = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
2220, 21eqeq12i 2752 . . . . . . . . . . 11 (𝐶 = 𝐷 ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
23 abbib 2808 . . . . . . . . . . 11 ({𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ↔ ∀𝑥((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
2422, 23bitri 275 . . . . . . . . . 10 (𝐶 = 𝐷 ↔ ∀𝑥((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
25 breq1 5150 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
26 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → (rank‘𝑥) = (rank‘𝑧))
2726sseq1d 4026 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ((rank‘𝑥) ⊆ (rank‘𝑦) ↔ (rank‘𝑧) ⊆ (rank‘𝑦)))
2827imbi2d 340 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
2928albidv 1917 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))))
3025, 29anbi12d 632 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
31 breq1 5150 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥𝐵𝑧𝐵))
3227imbi2d 340 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → ((𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
3332albidv 1917 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))
3431, 33anbi12d 632 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
3530, 34bibi12d 345 . . . . . . . . . . 11 (𝑥 = 𝑧 → (((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))) ↔ ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))))))
3635spvv 1993 . . . . . . . . . 10 (∀𝑥((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))) → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
3724, 36sylbi 217 . . . . . . . . 9 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) ↔ (𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦)))))
38 simpl 482 . . . . . . . . 9 ((𝑧𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐵)
3937, 38biimtrdi 253 . . . . . . . 8 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝑧𝐵))
4019, 39jcad 512 . . . . . . 7 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → (𝑧𝐴𝑧𝐵)))
41 ensym 9041 . . . . . . . 8 (𝑧𝐴𝐴𝑧)
42 entr 9044 . . . . . . . 8 ((𝐴𝑧𝑧𝐵) → 𝐴𝐵)
4341, 42sylan 580 . . . . . . 7 ((𝑧𝐴𝑧𝐵) → 𝐴𝐵)
4440, 43syl6 35 . . . . . 6 (𝐶 = 𝐷 → ((𝑧𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑧) ⊆ (rank‘𝑦))) → 𝐴𝐵))
4517, 44biimtrid 242 . . . . 5 (𝐶 = 𝐷 → ((𝑧 ∈ {𝑤𝑤𝐴} ∧ ∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦)) → 𝐴𝐵))
4645expd 415 . . . 4 (𝐶 = 𝐷 → (𝑧 ∈ {𝑤𝑤𝐴} → (∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴𝐵)))
4746rexlimdv 3150 . . 3 (𝐶 = 𝐷 → (∃𝑧 ∈ {𝑤𝑤𝐴}∀𝑦 ∈ {𝑤𝑤𝐴} (rank‘𝑧) ⊆ (rank‘𝑦) → 𝐴𝐵))
4811, 47mpi 20 . 2 (𝐶 = 𝐷𝐴𝐵)
49 enen2 9156 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
50 enen2 9156 . . . . . . 7 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
5150imbi1d 341 . . . . . 6 (𝐴𝐵 → ((𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
5251albidv 1917 . . . . 5 (𝐴𝐵 → (∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦))))
5349, 52anbi12d 632 . . . 4 (𝐴𝐵 → ((𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))) ↔ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))))
5453abbidv 2805 . . 3 (𝐴𝐵 → {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))})
5554, 20, 213eqtr4g 2799 . 2 (𝐴𝐵𝐶 = 𝐷)
5648, 55impbii 209 1 (𝐶 = 𝐷𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1534   = wceq 1536  wex 1775  wcel 2105  {cab 2711  wne 2937  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  wss 3962  c0 4338   class class class wbr 5147  cfv 6562  cen 8980  rankcrnk 9800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-r1 9801  df-rank 9802
This theorem is referenced by: (None)
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