Theorem fineqvomon 35378

Description: If all sets are finite, then the class of all natural numbers equals the proper class of all ordinal numbers. (Contributed by BTernaryTau, 30-Dec-2025.)

Assertion

Ref	Expression
fineqvomon	⊢ (Fin = V → ω = On)

Proof of Theorem fineqvomon

Step	Hyp	Ref	Expression
1		onfin2 9181	. 2 ⊢ ω = (On ∩ Fin)
2		ineq2 4166	. . 3 ⊢ (Fin = V → (On ∩ Fin) = (On ∩ V))
3		inv1 4351	. . 3 ⊢ (On ∩ V) = On
4	2, 3	eqtrdi 2812	. 2 ⊢ (Fin = V → (On ∩ Fin) = On)
5	1, 4	eqtrid 2808	1 ⊢ (Fin = V → ω = On)

Syntax hints:   → wi 4   = wceq 1559  Vcvv 3453   ∩ cin 3903  Oncon0 6342  ωcom 7842  Fincfn 8923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-om 7843  df-1o 8432  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927

This theorem is referenced by:  fineqvomonb  35379  fineqvr1ombregs  35398