unir1regs - Mathbox for BTernaryTau

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Theorem unir1regs 35310

Description: The cumulative hierarchy of sets covers the universe. This version of unir1 9737 replaces setind 9668 with setindregs 35305. (Contributed by BTernaryTau, 30-Dec-2025.)

**Assertion**
Ref	Expression
unir1regs	⊢ ∪ (𝑅₁ “ On) = V

**Proof of Theorem unir1regs**
Step	Hyp	Ref	Expression
1		setindregs 35305	. 2 ⊢ (∀𝑥(𝑥 ⊆ ∪ (𝑅₁ “ On) → 𝑥 ∈ ∪ (𝑅₁ “ On)) → ∪ (𝑅₁ “ On) = V)
2		vex 3446	. . . 4 ⊢ 𝑥 ∈ V
3	2	r1elss 9730	. . 3 ⊢ (𝑥 ∈ ∪ (𝑅₁ “ On) ↔ 𝑥 ⊆ ∪ (𝑅₁ “ On))
4	3	biimpri 228	. 2 ⊢ (𝑥 ⊆ ∪ (𝑅₁ “ On) → 𝑥 ∈ ∪ (𝑅₁ “ On))
5	1, 4	mpg 1799	1 ⊢ ∪ (𝑅₁ “ On) = V

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3442 ⊆ wss 3903 ∪ cuni 4865 “ cima 5635 Oncon0 6325 𝑅₁cr1 9686

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-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-regs 35301

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-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 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-iun 4950 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-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 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-ov 7371 df-om 7819 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-r1 9688

This theorem is referenced by: fineqvr1ombregs 35313

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